How does the preference for increasing payments depend on the size and source of the payments ?

It is well-known that subjects can exhibit a preference for increasing payments. Smith (2009a) makes a related prediction that the di¤erence between the preference increasing wage payments and the preference for increasing non-wage payments will be largest for intermediate payments. We nd evidence consistent with this prediction. Consistent with previous experiments, we nd that the preference for increasing payments is increasing in the size of the payments. Also consistent with the literature, we nd that the preference for increasing wage payments is stronger than the preference for non-wage payments.


Introduction
It is well-known that subjects often exhibit a preference for sequences of outcomes with an improving trend over sequences that are constant or declining (Loewenstein & Prelec, 1993).In fact, many subjects prefer increasing sequences of payments over constant sequences, even if the increasing sequences have a lower present value (Loewenstein & Sicherman, 1991).Loewenstein and Sicherman (1991) also found that the preference for increasing payments is stronger when the money is described as wages rather than from another source.As an explanation of the difference in the preference for increasing wage payments and non-wage payments, Smith (2009a) offered a model of a decision maker with imperfect memory who makes a prospective choice among payment sequences.Smith (2009a) predicts that the difference between the preference for increasing wage payments and the preference for increasing non-wage payments will be largest for intermediate amounts.
In order to test the prediction of Smith (2009a), Duffy and Smith (2013) offered subjects a series of questions where each response item specified an explicit sequence of payments over time.Within each question, there was an option for a constant payment sequence.We refer to this constant amount as the base amount of the question.The other response items were increasing sequences of payments, where each response item varied in its rate of increase.Within each question, the undiscounted sum of each payment stream was identical among all response items.Therefore, the rate of the increase of the sequence is negatively related to the present value of that sequence.Additionally, the payments were either describes as wages or non-wage money.We interpret the former as requiring the expenditure of costly effort, whereas the latter does not.Duffy and Smith (2013) found that the preference for increasing sequences of income is stronger when the payments are larger and described as payments from wages as opposed to payments of non-wage money.The authors also found some evidence that the difference between the preferences for increasing wage payments and the preference for increasing non-wage payments was largest for intermediate payments.The authors interpreted this evidence as consistent with the prediction of Smith (2009a).
However, the design of Duffy and Smith (2013) exhibits a shortcoming: the prediction of Smith (2009a) was tested by employing sequences that were constructed using a proportional technique, whereby the amounts were obtained by multiplying the base amounts by fixed proportions.This technique implies that questions with a larger base amount exhibited a greater amount of increases when compared to questions with a smaller base amount.It is therefore possible that subjects were attracted to the greater increasing amounts and that prediction of Smith (2009a) would not be supported by choice among sequences constructed by a different technique.Duffy and Smith (2013) also elicited preferences over sequences that were constructed by an additive technique, which adds fixed amounts to the base amounts.The authors found that the relationship between the preference of increasing payments and the size of the payments was not different between the items constructed with the additive and proportional techniques.However, in these elicitations that used the additive technique, the description of the source of the money was not varied and therefore the predictions of Smith (2009a) have not been sufficiently scrutinized.
In this paper, we construct sequences of payments via the additive technique, and explore the effect of changing the base amounts and the description of the sources of the payments.Similar to Loewenstein and Sicherman (1991), we find that the preference for increasing wage payments is stronger than the preference for non-wage payments.Additionally, consistent with Duffy and Smith (2013), we find that the preference for increasing payments is increasing in the size of the payments.Finally, even with the additive technique for constructing the sequences of payments, we find evidence in support of the predictions of Smith (2009a): the largest difference between the preference for increasing wage payments and the preference for increasing non-wage payments occurs for intermediate amounts.These results improve our understanding of the preference for sequences of money and can help guide the sequencing of wage payments.

Related Literature
There is an extensive literature that examines the preference for improving sequences of outcomes, which extends to monetary outcomes or nonmonetary outcomes, retrospective evaluations or prospective evaluations, short or long time horizons and even includes non-human subjects. 1We contribute to this literature in that we investigate the effects of the size of the payments and the source of the payments on the preference for increasing payments.
Similar to Loewenstein and Sicherman (1991), we find that the preference for increasing payments are particularly pronounced when the payments are described as "income from wages" as opposed to money from another source, which the authors describe as "income from rent."Here we do not utilize the "income from rent" description because if the subject has prosocial preferences, the subject might not want to obtain an improving sequence of money by imposing a declining sequence on the person paying the rent.We measure the preference for increasing payments of non-wage money by describing the payments as resulting from a large lotto jackpot won by a family member. 2Smith (2009a) presents a model of a decision maker with imperfect memory who makes a choice involving payment sequences in exchange for work-related effort.It is assumed that the decision maker has an uncertain cost of effort, and before the decisions regarding effort, the decision maker receives information about the cost of effort.After the action related to effort, the decision maker forgets the signal but makes an inference of its content from the objective features of the decision that are not forgotten: the wage paid and the choice of effort.Smith (2009a) shows that increasing payments imply a lower perceived cost of effort and thus a larger experienced surplus when compared to a constant payment profile.Intuitively, this is the case because a lower payment before the first choice of effort serves to reduce the perceived cost of effort in the following period.Here we find evidence consistent with this prediction: the difference between the preference for increasing wages and the preference for increasing non-wage payments is largest for intermediate amounts.
There is a strand of research that studies the relationships between the size of a single monetary payment, the delay in which it is received, and the subject's time preference. 3Here we perform a similar exercise in the sense that we wish to learn how the subject's time preferences (or negative time preference in our case) varies with the size of the payments.However, other than Duffy and Smith (2013), to our knowledge, there has not been a study that examines the relationship between the preference for increasing payments and the size of the payments.
There are two primary criticisms of the preference for increasing payments literature.First, there is evidence that the preference for increasing payments is not robust to the method of elicitation.Second, the responses of the subjects are not incentivized and should therefore be interpreted with caution.We now address these two criticisms.Frederick and Loewenstein (2008) show that the preference for improving sequences is sensitive to the means of elicitation. 4We design our questions in order to mitigate the spurious effects discussed by Frederick and Loewenstein.The authors list three reasons why a subject might exhibit a preference for improving sequences: the utility of anticipating future outcomes, a contrast effect by having a series of improvements according to a reference point, and an extrapolation effect where subjects come to believe that the payment trajectory will continue beyond that specified by the experimenter.These first two reasons are not driven by the means of elicitation, however we view the final reason to be an unwanted remnant of the methodology.Therefore, our experiment is designed to mitigate the extrapolation effect by explicitly stating that the choice of option will not affect subsequent income.
The other criticism is that the experimental work on the preference for increasing payments is largely not incentivized.Nonetheless, there is evidence that data generated by such experiments are useful and consistent with the empirical evidence.For instance, Johnson and Bickel (2002) do not find significant differences between the measurement of time preferences involving hypothetical and actual money.Additionally, a large body of empirical evidence supports the claim that people prefer increasing sequences.For instance, research finds that wages increase at a faster rate than productivity. 5It is difficult to see how this could persist unless the workers had a preference for such improvements.Finally, researchers find that happiness and satisfaction are related to increases in wages. 6Whereas we acknowledge the unincentivized nature of our study, we also note that the evidence regarding the preference for increasing payments includes unincentivized experiments and incentivized empirical work.

Experimental Procedure
A total of 398 undergraduate and graduate students in the psychology subject pool at Rutgers University-Camden were recruited to participate in the experiment.The subjects were given course credit for participating.Each response was entered on paper.
Subjects were each presented with five payment stream questions, which involved six possible payment sequences over six years.The subjects were told to select the one that they most prefer.In each of the five payment questions, the subject was presented with a constant sequence with a base amount of $17,000, $37,000, $57,000, $77,000, or $97,000.The other response items within each question varied the degree to which the payments were increasing.Therefore, we can associate each income question with the base amount of the sequence.These payment sequences were designed so that, within each question, each sequence option summed to an identical amount.Therefore, a subject who discounts in the standard fashion would select the constant sequence of income, regardless of the size of the payments.Further, within each question, the response items had identical values in the third year.However, the increasing response items each had lower payments in the first and second years, and higher payments in the fourth, fifth, and sixth years.Each sequence was constructed using the same additive technique where fixed amounts were added to the base amounts.See the appendix for a sample payment question and an explanation of this additive technique.
Additionally, we varied the order in which the questions were presented to the subjects.Also, we presented the response items so that they were ordered by their rate of increase.Approximately half of the subjects were given the options in ascending order: the constant sequence as the first option and the most increasing sequence as the last option.Approximately half were given the options in descending order: the most increasing sequence as the first option and the constant sequence as the last option.In the analysis of the data, we recoded the responses so that Option 1 represents the constant sequence and Option 6 represents the most increasing sequence.Since the rate of increase in the payments is negatively related to the present value of the sequence, and since we recoded the responses, we are therefore able to speak of a stronger preference for increasing payments as being associated with a higher option number.
In order to minimize the extrapolation effect (Frederick & Loewenstein, 2008), each response item included the description "same for each" for "year 7 and beyond."The subjects were told that the dollar amounts were listed in 2012 dollars and that their forecast of inflation should not be factored into their responses.Subjects were randomly selected to be in one of two treatments: the Job treatment or the Lotto treatment.Subjects in both treatments were given the identical five payment questions, however the description of the source of the money was different.Lotto treatment subjects were told that a relative won a substantial lotto jackpot and offered the following streams of money.Job treatment subjects were told that the payments are associated with wages from employment.
Finally, we also asked for their descriptive rating of the base amounts.Specifically, the subjects were asked to provide their descriptive rating of "starting salaries" of $17,000, $37,000, $57,000, $77,000, and $97,000 on a scale of 1 (very low) to 7 (very high).We excluded 15 subjects that did not complete every item and subjects that offered a non-monotonic descriptive rating.Of the 384 subjects, 183 were in the Lotto treatment and 201 were in the Job treatment.

Results
First, we investigate whether the preference for increasing payments is increasing in the size of the payments and whether the preference for increasing payments is stronger for payments described as wages.We perform a series of repeated measures regressions with the dependent variable as the payment option choice.The independent variables include the base amount of the question, a dummy variable indicating whether the question entailed wage or non-wage income, and an interaction between the two.Since we have repeated observations, we employ a repeated measures regression with an unstructured covariance matrix.In other words, we assume a unique correlation between any two observations involving a particular subject.However, we assume that observations involving two different subjects are statistically independent.7This analysis is summarized in Table 1.Consistent with the findings of Loewenstein and Sicherman (1991), we find that the preference for increasing payments is stronger when the income is described as income from wages as opposed to another source.Further, consistent with Duffy and Smith (2013), we find that the preference for increasing payments is stronger for larger amounts.However, we do not find that the interaction between the variables is significantly related to the preference for increasing payments.8 Now we investigate whether the difference between the preference for increasing wage payments and the preference for increasing non-wage payments is largest for intermediate payments, as predicted by Smith (2009a).In order to study this difference, we perform a t-test between the Lotto and Job treatments for each of the five income questions.We also perform a Mann-Whitney test between the Lotto and Job treatments for each of the five income questions.In addition to the means and standard deviations within each treatment, the results of the t-tests and Mann-Whitney tests are listed below in Table 2.We also perform a regression similar to that summarized in regression (3) in Table 1, with the exception that we treat the Base variable as categorical rather than continuous.We then perform a simple effects analysis, which tests for differences between the Job and Lotto treatments within each question.9This analysis is summarized in Table 2.All three analyses provide some support for the prediction of Smith (2009a) that the difference between the preference for increasing payments of wages and the preference for increasing payments of nonwage money is greatest for intermediate amounts.In particular, we see that the t-statistic, the Mann-Whitney z-statistic and the simple effects F-statistic are all largest for the intermediate $57,000 question.10

Discussion and Conclusion
We have investigated the nature of the preference for increasing payments.We elicited preferences over sequences of payments by varying the amount of the payment.We also varied the description of the source of the payments as either resulting from a job or from a lotto prize.Consistent with the findings of Loewenstein and Sicherman (1991), we find that the preference for increasing payments is stronger when the payments are described as income from wages rather than from another source.Further, consistent with the findings of Duffy and Smith (2013), we find that the preference for increasing payments is increasing in the size of the payments.
Using our data, we test a prediction of Smith (2009a), which states that the difference between the preference for increasing wage payments and the preference for increasing non-wage payments will be largest for intermediate payments.Smith (2009a) presents a model of a decision maker with an imperfect recall of the cost of effort where increasing wage payments can reduce the perceived cost of effort.For payments that are very likely or very unlikely to cover the cost of effort, the benefit of such a reduction is minimal.However, for payments that are neither likely nor unlikely to cover the cost of effort, there could be a significant benefit from such a reduction.Therefore, Smith (2009a) predicts that the difference between the preference for increasing wage payments and the preference for increasing non-wage payments will be largest for intermediate amounts.Duffy and Smith (2013) found evidence in support of this however, the construction of the increasing sequences and the relatively smaller sample size rendered the evidence to be less than fully satisfactory.Here we study the question with an improved construction of the increasing sequences and a larger sample size and we find evidence in support of the prediction of Smith (2009a).We hope that our results contribute to a better understanding of the preference for sequences of payments, in general, and the optimal sequencing of wages, in particular.