Voluntary contributions in a system with uncertain returns: a case of systemic risk

Abstract

This paper investigates systemic risk that emerges from the interplay between uncertain returns to individual actions, uncertainty on others’ behavior and all this filtered through individual attitudes toward risk. We design a finitely repeated linear public good experiment based on a voluntary contribution mechanism and analyze the effect of risky and uncertain returns on subjects’ contributions. Results from a baseline treatment without uncertainty are compared with two risky treatments characterized by different values for the marginal per capita return. In the treatments with risk, subjects are randomly assigned to one out of three feasible marginal per capita returns, independently of what their individual contribution was. Results show that a sufficient level of uncertainty leads to significantly lower individual contributions. Furthermore, in a system with lower contributions due to uncertainty, subjects’ risk aversion enhances the systemic risk, leading the system to collapse.

Appendix A: Instructions to subjects (translated from Spanish)

Thank you very much for being here. The instructions are identical to all participants. Read them carefully. If you have any questions or concerns, please raise your hand and we will answer your questions individually. During the session, it is strictly forbidden to communicate with the other participants. The unit of experimental money will be the ECU (Experimental Currency Unit), where $100~\hbox {ECU} = 10~\hbox {Euro}$. At the end of the session, one of your decisions will be randomly chosen. Note that all choices are equally likely. The experimental payoff corresponding to the selected decision will be calculated, converted to Euro, and paid to you (privately) in cash.

The Experiment (Baseline (BL) treatment)

The experiment consists of 10 independent periods, in which you will interact with 2 other participants in the session. The 3 of you form a group that will remain THE SAME in all periods. The identity of the participants of your group will not be revealed to you at all during the whole session. At the beginning of each period, each participant receives an endowment of 100 ECU. In any period, each member of the group has a decision to make.

Every period, you have to decide how much of your endowment you want to contribute to a common project. Your contribution will be an amount not smaller than 0 ECU and not greater than 100 ECU. Furthermore, it must be an integer number. You will keep for yourself any amount of your endowment that you decide not to invest in the common project (“ECU you keep”).

In every period, your earnings consist of two parts:

• 1. the “ECU you keep” $= [100 - \hbox {your contribution}]$ ECU;

• 2. the “income from the project.”

The “income from the project” is calculated by adding up the contributions of the 3 members of your group and multiplying the resulting sum by a number that we call $\alpha$. That is:

Income from the project $= [\hbox {Your contribution} + \hbox {Your partners' contribution}] \times \alpha$

The multiplier $\alpha$ is equal to 0.6. [Treatment HR] The multiplier $\alpha$ can be either 0.9 or 0.6 or 0.3, [Treatment LR]: The multiplier $\alpha$ can be either 0.75 or 0.6 or 0.45, where each value is equally likely. You have to decide about your contribution without knowing the value of $\alpha$.

The income from the common project is determined in the same way for every member of the group, and it is independent of the size of your individual contributions.

[Treatments HR and LR] The ”income from the project” is determined as follows: each member is randomly assigned a number ranged between 1 and 3 regardless of the size of his/her contribution. Given that several (or all of them) members of the group can be assigned the same number, this is not necessarily a ranking. In case the extracted number is different for each subject, the subject with number 1 receives $\alpha =0.9$ [Treatment LR: $\alpha = 0.75$] from the common project; the subject with number 2 receives $\alpha =0.6$, and the subject with number 3 receives $\alpha = 0.3$ [Treatment LR: $\alpha =0.45$]. If it is the case that several (or even all subjects) subjects get the same number, ties are solved as follows:

• If all three members get the same number (1, 2 or 3), each member receives $\alpha = (0.9 + 0.6 + 0.3)/3 = 0.6$ [Treatment LR: $\alpha = (0.75 + 0.6 + 0.45)/3 = 0.6$];

• If two members get number 1 or 2, they both receive $\alpha = (0.9 + 0.6 )/2 = 0.75$ [Treatment LR: $\alpha = (0.75 + 0.6 )/2 = 0.675$] and the third member receives $\alpha = 0.3$ [Treatment LR: $\alpha = 0.45$];

• If two members get number 2 or 3, they both receive $\alpha =(0.6 + 0.3)/2 = 0.45$ [Treatment LR: $\alpha = (0.6 + 0.45)/2$] and the first ranked member receives $\alpha$ = 0.9 [Treatment LR: $\alpha =0.75$];

[Treatment BL] Example: If the sum of the contributions of the three members is 60 ECU, each member receives an income from the project equal to $(0.6 \times 60) = 36$ ECU.

[Treatment HR] Example: If the sum of the contributions of the three members is 60 ECU and each member is assigned a different number, the subject with number 1 receives an income from the project of $(0.9 \times 60) = 54$ ECU [Treatment LR: $(0.75 \times 60) = 45$ ECU]; the one with number 2 receives $(0.6 \times 60) = 36$ ECU and the subject with number 3 receives $(0.3 \times 60) = 18$ ECU [Treatment LR: $(0.45 \times 60)=27$ ECU]. However, for instance:

• if all members are assigned the same number (1, 2 or 3), each receives $(0.6 \times 60) = 36$ ECU;

• if two members are assigned the same number (1 or 2), they each receive $[(0.9 + 0.6)/2 \times 60] = 45$ ECU [Treatment LR: $[(0.75 + 0.6)/2 \times 60] = 0.67 \times 60= 40.2$ ECU]; the other subject receives $(0.3 \times 60) = 18$ ECU [Treatment LR: $(0.45 \times 60) = 27$ ECU];

• if two members are assigned the same number (2 or 3), they each receive $[(0.6 + 0.3)/2 \times 60] = 27$ ECU [Treatment LR: $[(0.6+ 0.45)/2 \times 60] = 0.525 \times 60 = 31.5$ ECU]; the other member of the group receives $(0.9 \times 60) = 54$ ECU [Treatment LR: $(0.75 \times 60) = 45$ ECU].

At the end of each period you will receive information about the individual contribution of your partners in the group and your own period-earnings. Before the experiment starts, you will have to answer some control questions to verify your understanding of the rules of the experiment. Please remain seated quietly until the experiment starts. If you have any questions please raise your hand.