A new test of convexity–concavity of discount function


Discounted utility theory and its generalizations (e.g., quasihyperbolic discounting, generalized hyperbolic discounting) use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity–concavity of discount function. This test can be used with any utility function (which can be linear or not) and any preferences over risky lotteries (expected utility theory or not). The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function (for delays up to two month).

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Fig. 1


  1. 1.

    Stationarity is a behavioral axiom that can be directly tested on empirical data since it is formulated in terms of revealed preferences. Convexity of discount function is a property of an unobserved theoretical function that cannot be directly tested on empirical data. From this point of view, the contribution of our paper is to formulate the convexity property (Jensen’s inequality) in terms of revealed preferences so that it can be directly tested on empirical data.

  2. 2.

    The hyperbolic factor method of Rohde (2010) and time-tradeoff method of Attema et al. (2010) share the same drawback (complex incentive-compatible elicitation of exact indifferences).

  3. 3.

    The distractor task was cognitive reflection test (Frederick, 2005) consisting of three questions.

  4. 4.

    Wilcoxon signed-rank tests were used as normality of distributions was not supported by the data.

  5. 5.

    An alternative \(L\equiv ({l}_{0},{l}_{1},{l}_{2})\) second-order dominates an alternative \(R\equiv ({r}_{0},{r}_{1},{r}_{2})\) when \({l}_{0}+{l}_{1}+{l}_{2}\ge {r}_{0}+{r}_{1}+{r}_{2}\) as well as \({l}_{0}\ge {r}_{0}\), \({2l}_{0}+{l}_{1}\ge 2{r}_{0}+{r}_{1}\) and \(3{l}_{0}+{2l}_{1}+{l}_{2}\ge {3r}_{0}+{2r}_{1}+{r}_{2}\) and at least one of these inequalities holds as a strict inequality.


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Correspondence to Pavlo R. Blavatskyy.

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Pavlo Blavatskyy is a member of the Entrepreneurship and Innovation Chair, which is part of LabEx Entrepreneurship (University of Montpellier, France) and funded by the French government (Labex Entreprendre, ANR-10-Labex-11-01).


Appendix A

See Table 2.

Table 2 Number (%) of subjects revealing each of nine possible choice patterns

Appendix B


Thank you for participating in this experiment. The experiment is designed to study individual decisions. You will make your choice on the computer. The computer will communicate your gains at the end of the experiment. Your decisions in the experiment are private. Please do not communicate with others during the experience.

The instructions are simple, if you follow them carefully, you could earn a considerable amount of money. Your winnings will depend on the decisions you make.

The experiment is divided into three parts and a questionnaire:

  1. 1.

    In a first part, we will ask you to choose between two options, and this several times. Each option offers you a given amount of money at three different delays.

  2. 2.

    In a second part, we will ask you respond to three questions of logic.

  3. 3.

    In a third part, we will ask you to choose between two options several times. Each option offers you a given amount of money at three different delays.

  4. 4.

    Finally, you will answer questions about your personal situation.


At the end of the experiment, one question of Parts 1 or 3 will be randomly selected and played for real. Part 2 will offer you a bonus per correct answer.

Detailed instructions

  • Part 1: In this part, you will choose between two options that offer you three amounts of money at three different dates: an amount today, an amount in one month and an amount in 2 months.



    In this question, you will make a choice between two options: a right option and a left option. You save your choices on the computer. Click on the option you prefer. Your preferred option will be highlighted in red, and then click on the bottom OK to confirm your choice.

    For this example, if you choose the left option, you will receive nothing today, but you will receive €20 in one month and €10 in two months. If you prefer the right option, you will receive €10 today, €10 in one month and €10 in two months.

    In this first part, you are asked to make 100 choices similar to the example above.

    To avoid you to come back to receive your future earnings, the amounts you will receive in one month and/or in two months will be deposited in your internal mailbox on the scheduled date or will be sent to you to another address (if you prefer) so that they arrive in your mailboxes on the scheduled date. Please inform the experimenter of what suits you for future payments.

    The following text was not printed on the instructions, but was read by the experimenter:

    On the right side of your computer, there are two envelopes: one white and one brown. You will receive the white envelope with your earlier payment in one month and the brown envelope with the later payment in two months. Inside each envelope, there is a letter reminding you your participation in the experiment one (two) month(s) ago and your payment. After the payment information at the end of the experiment, you have to address the envelope(s) to yourselves and write your future payment(s) in the reminder letter of the corresponding envelope.

    • Part 2: In this part, you will answer 3 questions of logic. You will receive 0.50 € per correct answer.

    • Part 3: In this part, you will make the same 100 choices similar of Part 1. You will choose between two options that offer you three amounts of money at three different dates: an amount today, an amount in one month and an amount in 2 months.

      Do you have any questions?

Appendix C

See Table 3 .

Table 3 Raw experimental data

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Blavatskyy, P.R., Maafi, H. A new test of convexity–concavity of discount function. Theory Decis 89, 121–136 (2020). https://doi.org/10.1007/s11238-020-09747-3

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  • Intertemporal choice
  • Time preference
  • Discounted utility
  • Convexity
  • Discount function