2.3.1 Production and cost functions

Workers are invited to make consecutive effort choices for a number of piece rates. Their income is composed of a fixed share, 90 ECU (9 €), and a variable share. The variable share is the product of their effort level ( $e_i$ ), their marginal productivity $\left({\alpha }_i,\in \{,{\alpha }_H,,,{\alpha }_L\right)$ ), and a piece rate ( $w_i$ ) minus the cost for a given level of effort $\left(c,\left(e_i\right)\right)$ . As we will explain in more detail below, workers are free to choose an effort level, while they are informed that an (anonymous) principal will choose their piece rate.

${\alpha }_i$ is allocated according to the subjects' performance at an aptitude test that the workers take after receiving the instructions about the workplace setting described above, and after completing a comprehension test.Footnote ⁵ They are informed that performing better at the aptitude test will increase their chances of having higher productivity (a high $\alpha$ ). Using an aptitude test to generate heterogeneity in productivity across agents in a stated effort experiment has been used in gift-exchange experiments to justify induced productivity differences (Bolton & Werner, Reference Bolton and Werner2016; Gross et al., Reference Gross, Guo and Charness2015). The idea is to overcome a certain arbitrariness in productivity differences by creating a link between induced and real ability that would not exist under random ability allocation. Furthermore, we deliberately use an aptitude test that not only accounts for innate ability or willingness to work hard, but that may also depend on the education that the worker benefited from. We made this design choice to capture the fact that principals often deal with agents that have different ability levels precisely because they benefited from different education levels. This will then also be relevant for their evaluation of incentives schemes from a fairness perspective.

The aptitude test consists of nine questions: three logic questions, three French questions and three general knowledge questions. The French and logic questions were simplified versions of TAGE MAGE, a French equivalent of GMAT. Workers have 10 min to complete a practice test (same format but different questions) and then have 5 min to complete a test that will define their production function.Footnote ⁶ Ability is determined at the pair level. We assign ${\alpha }_H$ to the worker with the best performance within the pair and ${\alpha }_L$ to the other one and in all sessions we define ${\alpha }_H=60$ and ${\alpha }_L=40$ .Footnote ⁷

The cost function is constant across agents, and it is convex in effort choices. Figure A1 in the appendix displays the production and cost function of both workers.

2.3.2 Workers’ decisions

The agents make effort choices for all piece rates that can be chosen by the principal. As is common in the strategy method, they are informed that the principal will only choose one of their choices as payoff-relevant.Footnote ⁸

Piece rates range from 0.30 to 0.70 ECU (for high-ability workers) and from 0.30 to 0.75 ECU (for low-ability workers) in increments of 0.05. It is possible that workers will react differently to a certain wage if the previous piece rate was higher or lower. Nonetheless, we decided not to completely randomize the order applied to the workers because it is unfeasible to robustly identify order effects under complete randomization (81 possible combinations would need to be compared). However, we test for order effects by looking at two benchmark cases: (1) ascending order of piece rates starting at 0.30 and ending at 0.70 ECU; and (2) descending where the order is reversed. One of the order is randomly assigned to each worker.

Workers choose effort levels from a discrete set between 0 and 5 ( $e\in \{0,0.5,1,\cdots ,5\}$ ). We elicit effort decisions for all piece rates. The final income of the worker is $y_i=w_i{\alpha }_i{e}_i-c\left(e_i\right)+90$ ECU, where $c\left(·\right)$ is the effort cost function.

A screenshot of agent B's decision can be found in the appendix, Figure A2. For each piece rate, workers can view their production table showing how each effort level translates to production, effort cost and net variable income. To ease the cognitive burden, we show them a simulation of the consequences of their decision when clicking on a particular effort level. For instance, when effort level 3 is selected, the screen shows the worker’s production output (180 units), the current piece rate (0.5 ECU), the cost (48 ECU) and the net income (42 ECU) associated with such an effort level.

2.3.3 Workers’ information set

Workers are informed that the payoff-relevant piece rate will be chosen by a principal but they are not informed that this principal also chooses a piece rate for another worker. We chose this feature of the design to avoid horizontal wage comparisons among workers that could lead them to sabotage very unequal piece rates on the basis of their own fairness motives. This design feature is vital because we want to focus on the principals' reaction to wage inequality among workers; thus, we want to eliminate other, possibly confusing, factors from the principal's decision, as far as possible.Footnote ⁹

Furthermore, workers are not informed how their decisions affect the principal in order to avoid vertical social preferences that have been documented in the field (Ashraf & Bandiera, Reference Ashraf and Bandiera2018; DellaVigna et al., Reference DellaVigna, List, Malmendier and Rao2016). As explained below, in the Stakeholder group agents' actions affect the principals' income. Informing agents how their actions may or may not affect the income of the principal creates an additional uncontrolled source of variation across treatment groups. It would lead to treatment differences in principals' beliefs about the workers' behavior. We shut down this mechanism by restricting the information set of workers.

2.4.1 Belief elicitation

Principals are invited to choose wages for the pair of workers they are matched with. Prior to making these decisions, we elicit their beliefs about the effort level chosen by the workers for each piece rate they could possibly implement. We elicit beliefs regarding each worker’s effort sequentially to avoid asking too many questions at once. The workers’ order of appearance (either Worker A or Worker B) is randomized at the principal level. At the end of the experiment, we randomly draw one guessed belief, and if the principal's guess is correct she receives 10 ECU (1€).Footnote ¹² The drawing of the payoff-relevant piece rate in the belief elicitation is completely independent of the drawing of the payoff-relevant choice in the latter part of the experiment in order to achieve independence in the decisions across the two parts.

Belief elicitation of workers' effort choices plays a vital part in the experiment. It enables us to determine whether an egalitarian contract choice originates from normative distributive preferences or different beliefs regarding how workers should behave under each contract. Principals may believe that workers do not seek to maximize their own income and would choose different effort levels instead of the best responses. Under such a belief structure, an egalitarian contract may become optimal. In other words, eliciting beliefs enables us to determine whether our classification contracts (the extent of output and inequality they produce) is also shared by principals or not.

2.4.2 Contract decisions

After the belief elicitation part of the experiment, principals make 16 binary decisions between two contracts, where each contract consists of two piece rates (one for the more productive worker $Worke{r}_H$ and one for the less productive worker $Worke{r}_L$ ). The choices are summarized in Table 1, showing the piece rates associated with each decision, as well as the distributive and productive consequences of each option (conditional on the workers best responding to the piece rate).

Table 1

Set of decisions made by the principal assuming workers' choose effort to maximize their own income

	Piece rates		Income contract 1		Income contract 2		Joint output
	Contract 1	Contract 2	Worker H	Worker L	Worker H	Worker L	Contract 1	Contract 2	$\Delta$ -output
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Egalitarian versus high-inequality
Nr
1	0.4–0.6	0.55–0.45	25.5	25.5	51.5	13.5	250	270	20
2	0.4–0.6	0.6–0.4	25.5	25.5	63	10.5	250	300	50
3	0.5–0.65	0.55–0.45	42	30.5	51.5	13.5	280	270	−10
4	0.5–0.65	0.6–0.4	42	30.5	63	10.5	280	300	20
5	0.5–0.65	0.65–0.35	42	30.5	75.5	8	280	310	30
6	0.5–0.65	0.7–0.3	42	30.5	90	6	280	340	60
7	0.5–0.75	0.6–0.4	42	42	63	10.5	300	300	0
8	0.5–0.75	0.65–0.35	42	42	75.5	8	300	310	10
9	0.5–0.75	0.7–0.3	42	42	90	6	300	340	40
Equal piece rate versus high-inequality
10	0.5–0.5	0.55–0.45	42	17	51.5	13.5	260	270	10
11	0.5–0.5	0.6–0.4	42	17	63	10.5	260	300	40
12	0.55–0.55	0.6–0.4	51	20.5	63	10.5	290	300	10
13	0.55–0.55	0.65–0.35	51	20.5	75.5	8	290	310	20
14	0.55–0.55	0.7–0.3	51	20.5	90	6	290	340	50
Egalitarian versus equal piece rate and high-inequality
15	0.4–0.6	0.5–0.5	25.5	25.5	42	17	250	260	10
16	0.5–0.65	0.55–0.55	42	30.5	51	20.5	280	290	10

This table shows the series of decisions principals are asked to make. All units in columns (1)–(6) are in ECUs. The units in columns (7)–(9) are production quantities. The first two columns display the piece rates that are associated with each contract. The left-hand piece rate is the piece rate for the most productive worker (Worker _H) and right-hand piece rate is for the least productive worker (Worker _L) of the pair. The decisions can be split into egalitarian vs high-inequality and equal piece rate vs high-inequality choices. Egalitarian contracts result in outcomes proportional to effort. The equal piece rate contracts result in outcomes proportional to output High-inequality contracts maximize output in most of the cases. Columns (3)–(6) correspond to the variable share of income and thus exclude the 90 ECU show-up fee, common to all workers. The variable income levels (columns (3)–(6)) and the joint output for each contract (columns (7)–(8)) are conditional on the workers best responding to the piece rate

Before detailing the choices that principals face, we want to clarify the different types of contracts. Our experiment studies three types of contracts. First, we have egalitarian contracts, where the low ability worker ( $Worke{r}_L$ ) receives a higher piece rate than the high ability worker ( $Worke{r}_H$ ). If workers best respond to wages, the egalitarian piece rate will either perfectly equalize income levels (as in Contract 1 of Choices 1–2, 7–9, and 15) or significantly decrease inequality compared to its alternative (as in Contract 1 of Choices 3–7 and 16) because the low ability worker now receives more for each unit produced. This results in a situation where equality in effort yields equality in income. Second, we have equal piece rate contracts. These contracts pay an equal piece rate to each worker. They, thus, yield a situation where equality in production leads to equality in incomes. However, in this case, equal effort does not yield equal income, since the two workers differ in their productivity. Third, we have a class of choices named high-inequality contracts, where $Worke{r}_H$ receives a higher piece rate than $Worke{r}_L$ . This contract leads to high inequalities ex-post but also to higher levels of joint output compared to the alternative.

In the choices that principals make, two of these contracts are posited against each other. As shown in Table 1, we have three classes of choices. First, we have choices, where principals choose between an egalitarian ( $Worke{r}_L$ receives a higher piece rate compared to $Worke{r}_H$ ) and a high-inequality (output-maximizing) contract. Second, we have choices, where principals choose between an equal piece rate contract (both workers receive the same piece rate) and a high-inequality contract. Third, we have choices where the egalitarian contract is posited against an equal piece rate contract. Contract 2 yields higher output compared to Contract 1 in all decisions, except for Choice 3, where the egalitarian choice is also output-maximizing. This Choice permits us to test for situations in which the egalitarian or equal piece rate contract is output-maximizing to avoid positing that equality is always desirable. Some people may consider that ability-induced inequality is fair. However, Contract 2 always leads to larger inequality when workers best respond, hence its labeling “high-inequality contract”.Footnote ¹³

Within each class, choices were calibrated so that both inequality and joint output vary across choices, but without a perfect positive correlation. Otherwise, it would have been impossible to disentangle their respective impacts on contract choices. Figure 1 shows how differences in inequality between Contract 2 and 1 (on the y-axis) and output (difference in output between Contracts 1 and 2, on the x-axis) vary across choices. Orange dots represent each case in which Contract 1 is egalitarian and blue dots show when Contract 1 is an equal piece rate (equal procedure) contract. Choices with an egalitarian contract are naturally located at the top of the graph since they lead to a more drastic compression of income than equal piece rate contracts. The difference in inequality ranges from 13 to 84 ECU and difference in output ranges from −10 to 60 units produced. In ECU terms, the difference in output-based income is twice as small, since each unit of output is sold at 0.5 ECU. Therefore, in ECU terms, we can say that the inequality level varies more than the output level across choices. This calibration decision is based on pilot data showing that if output differences are too large across Contracts 1 and 2, principals eventually adopt a corner solution in which they maximize income. Consequently, if inequality and output varied on about the same scale, we would not be able to see that people also care about inequality to some extent: all principals would be mistakenly described as selfish income-maximizers. In this study, we focus on the window in which there is a trade-off between maximizing output and equality.

Fig. 1

Contract trade-offs assuming best responses

Notes: The Figure plots the theoretical trade-offs (assuming best responses), underlying the 16 contract choices that principals have to make. The y-axis shows the difference in inequality between both contracts, and the inequality of a contract is measured by the high-ability worker’s wage minus the low-ability worker’s wage. Hence, Contract 2 becomes increasingly unequal relative to Contract 1 as we move up the y-axis. The x-axis is the difference in output between contracts. Contract 2 becomes more efficient relative to Contract 1 as we move to the right-hand side of the plot. Yellow dots represent the trade-offs of equal piece rate contracts vs high-inequality contracts, and the blue dots represent the trade-offs of egalitarian contracts vs high-inequality contracts

Figure A3 in the appendix shows how we asked principals to make contract choices during the experiment. The top part of the screen shows the tables summarizing the information for Workers A and B,Footnote ¹⁴ the middle part asks principals to choose between both contracts, and the bottom part simulates the consequences of such a choice, both for the workers and for the principal. The simulation helps to ease the cognitive burden and saves computation time. It is based on the effort belief elicited beforehand. We remind them of the effort level they expect their workers to choose. We then inform them about the expected production associated with such effort levels and the variable income that each worker would receive under the selected contract. The table is updated when the principal selects a different contract. We instruct them to try out both simulations before making a choice.

Since this screen must be repeated 16 times for each of the choices, we randomize several features to avoid any anchoring biases. The 16 choices appear in a random order at the subject-level. Within a choice, the labeling of contracts as “Contract 1” or “Contract 2” is randomized. This implies that people cannot always choose Contract 2 to maximize their own income. The “Worker A” and “Worker B” labels are randomly assigned to the high-ability and low-ability workers and are thus independent of productivity differences.

2.4.3 Treatments

Between subjects, we will implement two treatments: (1) the spectator treatment and (2) the stakeholder treatment . The treatment varies across sessions, meaning that principals in the same session faced the same treatment.

In the spectator treatment, the principal receives a fixed wage of 20€ that is completely independent of her workers' output. The treatment enables us to identify how principals trade-off output and distributive concerns without holding any monetary stake in workers' joint output themselves; it thus constitutes a normative benchmark. In each decision, the principal is asked to make a trade-off between the implementation of an egalitarian (or equal piece rate) and an high-inequality contract (which is output-maximizing in most cases), keeping her own income constant across all the decisions. The size of the trade-off is documented in column (9), if the principals believe the agents are best-responding. The treatment can also be seen as analogous to a situation in which principals have no personal stake in the outcome of their organization (e.g. most civil servants).

In the stakeholder treatment, the principal receives a fixed participation fee of 60 ECU (6 €) and a variable share from the sales of the output produced by the workers. For each unit produced, she receives 0.5 ECU. She now faces a trade-off between maximizing her own income and implementing an egalitarian (or equal piece rate) contract. By analyzing choice patterns, we can infer from this treatment the price the principals are willing to pay in order to implement an egalitarian or equal piece rate contract. The size of the trade-off depends largely on the principals' beliefs regarding whether or not they expect workers to best respond to the piece rates. This highlights the importance of the belief-elicitation part of the experiment.

2.4.4 Principals' information set

Principals have full information about the environment they face. Principals are instructed that workers are not informed about the salary, nor the existence, of the second worker. They also know that the workers are not explicitly informed about how the principal herself is incentivized. Importantly, they know which of the workers is of low and high ability. Furthermore, they are also informed of the cost that workers have to endure for a given level of effort. This information is necessary for managers to infer the distributive consequences of their actions. Principals are specifically instructed about this in the comprehension test (see Appendix C).

To derive the hypotheses that we aim to test through the experimental design, we use a simple theoretical framework that helps characterizing our predictions on choice behavior. To that end, we assume that principals have preferences over their own income, $y_p$ (which depends on firms’ level of output in the Stakeholder treatment only), and the distribution of income between workers. Principals may also care about the level of output $\pi$ for intrinsic reasons. They may believe that maximizing output is the manager's job and they have to behave in this way because they are placed into this position; even Spectators may then care about output. We characterize those preferences using a simple utility function that we assume linear in income and the weight that principals put on the distributive outcomes:

(1) $\text{E(U) = E}\left(\left(y_p+\gamma \pi \left(e_H\left(W_H\right),e_L\left(W_L\right)\right)\right)-\beta M\left(y^{\ast }-Y\right)\right),$

where $y_p$ is the income of the principal, $\gamma$ is the weight the principal puts on output maximization for intrinsic relative to extrinsic motives ( $\pi \left(e_H,\left(w_H\right),,,e_L,\left(w_L\right)\right)$ is the joint output of the workers), and $\beta$ is the weight that the principal puts on implementing her preferred distribution of income. To characterize such distributive concerns, we assume that her utility decreases if the implemented distribution of income $y$ diverges from her preferred distribution of income $y^{\ast }$ . This loss in utility is characterized by the loss function $M\left(·\right)$ , which is weakly convex in the difference of the two. Our experimental design covers two distinct types of distributive concerns: (1) equality in workers’ income, where $y^{\ast }=\left(y_h\left(e\left(w_h\right)\right),y_l\left(e\left(w_l\right)\right)\right)$ such that $y_h=y_l$ and (2) equality in opportunity, where workers’ piece rates are equal: $y^{\ast }=\left(y_h\left(e\left(w_h\right)\right),y_l\left(e\left(w_l\right)\right)\right)$ such that $w_h=w_l$ .

The binary choices in our experimental design force principals to trade-off the motive to maximize output and distributive concerns. If $\beta =0$ then principals do not assign any importance towards minimizing $M\left(·\right)$ and they are not willing to trade-off output to implement an alternative distribution of income. In this case, they will prefer the output-maximizing contract in all 16 decisions. If $\beta$ is very large, then the utility gain from minimizing the loss-function $M\left(·\right)$ will always be greater than choosing contracts that maximize joint output and principals choose the contract that minimizes $M\left(·\right)$ in all 16 decisions. Finally, if $\beta >0$ but not too large, principals are sensitive to the size of the trade-off between inequality and output maximization and they will choose an output-maximizing contract once it gets too costly to choose the contract minimizing $M\left(·\right)$ .

The parameter $\gamma$ reflects how much Spectators care about output relative to Stakeholders, keeping the other-regarding part of the function constant. Thus, it characterizes crowding out of distributive concerns in contract calibration through financial incentives. If $\gamma =0$ we have a situation where spectators put no weight on maximizing output and there is no normative appeal for maximizing output. They would, hence, always choose to minimize $M\left(·\right)$ and never choose the output-maximizing contract if it conflicts with $M\left(·\right)$ . If $\gamma >0$ , output maximization has an intrinsic value and spectators are willing to trade-off output maximizing and distributive concerns; if $\gamma =1$ intrinsic motives to maximize output are as important as extrinsic concerns.Footnote ¹⁵

Finally, our design sheds light on what constitutes $y^{\ast }$ by studying whether subjects are more or less willing to trade off output to minimize inequality in outcomes or procedural inequality. If subjects care more about procedural equality, they should be more willing to forgo output when facing a choice between a high-inequality and an equal piece rate contract. To be more precise, since the equal piece rate contract is also inequality minimizing when posited against a high inequality contract, a preference for procedural equality is identified as a higher willingness to choose an inequality minimizing contract when it features equal piece rates than piece rates that favor the low ability worker.

This yields the following hypotheses:

Hypothesis 1

Existence of distributive concerns If principals hold significant distributive concerns ( $\beta >0$ ), they will be willing to forgo output in order to minimize inequality in incomes or inequality in procedure among their agents. Hence, output-maximizing contracts are not always chosen.

Hypothesis 2

Crowding out of distributive concerns If extrinsic motives to maximize output crowd out distributive concerns in contract choice ( $\gamma <1$ ), then a larger share of principals will choose the high-inequality contract if they are in the Stakeholder condition compared to the Spectator condition.

Hypothesis 3

Trade-off between output maximization and distributive concerns If there exists an intensive-margin trade-off between minimizing inequality and maximizing output, principals will be more likely to choose an output maximizing contract, the more costly the inequality minimizing or equal piece rate contract gets in terms of output forgone. If $\gamma >0$ , such a trade-off exists for both Spectators and Stakeholders.

Hypothesis 4

Equality in procedure versus equality in outcomes If principals are more concerned about equality in procedure than equality in outcomes, they are more willing to forgo output to implement an equal piece rate contract compared to an egalitarian contract.

For a derivation of these hypotheses using a random utility model, see Appendix B.