Hyperbolic Discounting, the Sign Effect, and the Body Mass Index

Abstract

Analysis of a broad survey of Japanese adults confirms that time discounting relates to body weight, not only via impatience, but also via hyperbolic discounting, proxied by inclination toward procrastination, and the sign effect, where future negative payoffs are discounted at a lower rate than future positive payoffs. Body mass index is positively associated with survey responses indicative of impatience and hyperbolic discounting, and negatively associated with those indicative of the sign effect. A one-unit increase in the degree of procrastination is associated with a 2.81 percentage-point increase in the probability of being obese. Subjects exhibiting the sign effect show a 3.69 percentage-point lower probability of being obese and a 4.02 percentage-point higher probability of being underweight than those without the sign effect. These effects are substantial compared with the prevalence rates of the corresponding body mass status. Obesity and underweight thus result in part from the temporal decision biases.

The original article first appeared in Journal of Health Economics 29:268–284, 2010. A newly written addendum has been added to this book chapter.

Appendices

Appendices

1.1 A.1Multinomial Probit Estimation of the Body Mass Status Function

Theoretically, time discounting variables may be non-monotonically correlated with body mass because being underweight, with a BMI much lower than the optimal level, 22, might have equally detrimental effects on health in the long run as being obese. If body mass were non-monotonically correlated to the time discounting variables, linear relations between BMI and time discounting as assumed in the regressions in the text would not be appropriate. Further, in the probit regressions for obesity, the reference category of dependent variable OBESITY would be inappropriate, too, since it includes the underweight status which may be positively correlated with discount rates, etc., as is the obese status.

To show that such non-monotonic associations are not observed between body mass and each of the time discounting variables, we put the results of multinomial probit regressions in panels (a) and (b) of Table 12.14, wherein, with the constraint that the sum of the probabilities of being underweight (BMI < 18.5), normal (18.5 ≤ BMI < 25), degree-1 obese (25 ≤ BMI < 30),²⁷ and severely obese (BMI ≥ 30) should equal one, marginal correlations between each of the probabilities and time discounting variables are estimated simultaneously by using the full sample and controlling for other personal attributes.

Table 12.14 Multinomial probit regressions for body mass status

Panel (b) shows that, in the case of model (2), correlations between body mass and DEBTIMP, PROCR, and SIGN are all monotonic²⁸: DEBTIMP and PROCR are both negatively correlated with underweight and normal body weight, whereas positively correlated with degree-1 obesity and severe obesity; SIGN is positively associated with underweight, whereas negatively associated with degree-1 obesity and severe obesity. For model (1), likewise, SIGN is monotonically correlated with body mass. However, as for DISCRATE and HYPERBOL, we can find no stable patterns of correlations.²⁹

In sum, as far as our data are concerned, associations between body mass and each of the time discounting variables seem to be monotonic, and the regression models in the text could be considered as appropriate.

1.2 A.2Estimating with Corrected BMI Data

1.2.1 A.2.1Correcting for Self-reporting Biases

As discussed in the text, the self-reported BMI data in the JHS05 may well contain underreporting biases. To check the robustness of the main results, we re-conduct the analysis by correcting for self-reporting bias.

We roughly examine whether or not the JHS05 data contain self-reporting biases by comparing the by-age BMI distributions of our JHS05 data with those of the NSHN04 data (sample size: 7689) that are actually measured.³⁰ Tables A2(a) through A2(d) describe statistically the by-age body mass distributions in the NSHN04 and JHS05 data. Results of the comparison suggest that the BMI means, the obesity rates, and the severe obesity rates in the original JHS05 data may contain underreporting biases. Particularly in the case of females, the BMI means and the obesity rate in JHS05 are significantly lower than those in NSHN04. Consistent with this tendency, the sample SDs of the females’ BMI in JHS05 are significantly smaller than those in NSHN04. As for the male sample, although the bias is not as large as in the female sample, the obesity rates are smaller in JHS05. There is no significant difference between the rates of severe obesity among males in the two data sets. Although the prevalence rate of underweight in JHS05 is slightly lower than that in NSHN04, the difference does not seem to be large.

Based on these findings, we correct for the underreporting bias in the female sample by specifying, from the JHS05 BMI data in generations i (i = 20, 30, 40, 50, 60), quadratic functions ${f}_i(x)={a}_i{x}^2+{b}_ix+{c}_i,$ which transform self-reported BMI values $x\ge 22$ to corrected BMI values f _i (x).³¹ The coefficients a _i , b _i , and c _i are determined such that the function satisfies: (1) ${f}_i\left({x}_i^{*}\right)=25$; (2) ${f}_i\left({x}_i^{**}\right)=30$; and (3) ${f}_i(22)=22$, where x ^* _i and x ^* * _i represent the critical BMI values by which to define obesity and severe obesity for generation i, respectively, that equilibrate the prevalence rates of obesity and severe obesity across JHS05 and NSHN04. Conditions (1) and (2) ensure that the corrected BMI distribution generates the same obesity and severe obesity rates as those in the NSHN04. Condition (3) is the assumption that since a BMI of 22 is known to be the healthiest,³² people with BMI ≤ 22 could be regarded as having no incentive to underreport their weights or overreport their height, and hence, would have no tendency to underreport their BMI values. The corrected values of the female BMI for $x\ge 22$ are computed by using the quadratic functions obtained for the corresponding generations, whereas for x < 22, no adjustment is made as there seems to be no serious reporting bias. As for the male BMI data, similar adjustment is made except that we do not correct data for x > 30 since the prevalence rate of severe obesity in the JHS05 does not differ significantly from that in the NSHN04 (see Table 12.17). Tables 12.15, 12.16, 12.17, and 12.18 show that the correction eliminates, to a great extent, the underreporting bias in the mean and the SD of each body mass status in each generation.³³

Table 12.15 By-age BMI distributions: NSHN04, JHS05, and corrected data

Table 12.16 By-age obesity distributions: NSHN04, JHS05, and corrected data

Table 12.17 By-age severe obesity distributions: NSHN04, JHS05, and corrected data

Table 12.18 By-age underweight distributions: NSHN04 and JHS05

1.2.2 A.2.2Results with Corrected Data

Tables 12.19, 12.20, and 12.21 provide the estimation results with the corrected data. As a whole, the association reported in the text between body mass and each of the three time discounting variables is robust even when corrected for self-reporting bias. For example, across the original and corrected data sets, there are few differences in the signs and significance levels of the estimated coefficients in the BMI regressions (see Tables 12.8 and 12.19).

Table 12.19 BMI regressions with corrected data

Table 12.20 Obesity regressions with corrected data

Table 12.21 Severe obesity regressions with corrected data

As a result of the correction, however, there are marginal changes in the results of the obesity regressions. In particular, as seen from the comparison of Tables 12.10 and 12.20, the magnitudes of the coefficients of impatience variables, especially those of DISCRATE, and their associated t-values become greater, whereas the opposite is true for the coefficients of the sign effect. Provided that our procedure successfully corrects for the actual underreporting biases, these marginal changes could be considered as the results of association between underreporting behavior and the two time discounting variables. For example, if obese respondents with high discount rates are more likely to underreport their weight, true positive correlations between the probability of being obese and the discount rate will be underestimated in the uncorrected sample. Similarly, if obese people with the sign effect tend to underreport BMI, true negative correlation between obesity and the sign effect will be overestimated in the uncorrected sample.³⁴ These findings suggest the importance of further study on behavioral aspects of underreporting behavior to detect unbiased correlations between time discounting and body mass.

1.3 A.3BMI Regressions Without Controlling for Smoking

Table 12.22 reports the results of the BMI regressions without controlling for smoking. By comparing the results with those with controlling for smoking in Table 12.8, we first see that our results do not change substantially even without controlling for smoking. Secondly, however, the magnitudes of coefficients and the associated t-values for significant time discounting variables are smaller in the smoking-uncontrolled regressions (Table 12.22) than in the smoking-controlled regressions (Table 12.8), with the coefficient of DISCRATE in model (1) being the only exception. As discussed in footnote 19, this implies that correlations between BMI and time discounting are underestimated when smoking is notcontrolled for.

Table 12.22 OLS regressions of BMI without controlling for smoking

Addendum: Robustness and Related Research

This addendum has been newly written for this book chapter.

In this addendum, we review recent evidence that demonstrates the robustness of the results in the previous article (Ikeda et al. 2010) regarding the association between time discounting and body weight found using the 2005 JHS data.

Our results remain valid for the post-2005 waves of the JHS. Based on the 2010 wave data, i.e., those of the most recent survey in which all the data required for the present purpose are available, time discounting and other related attributes continue to differ between obese and non-obese individuals, as summarized in Table 12.23. Consistent with our previous results based on the 2005 wave data, the average obese respondent exhibited higher personal discount rates (DR1 though DR5), higher inclination toward debt holding and procrastination (PROC), and a lower tendency of the sign effect. Additionally, as in the previous study, hyperbolic discounting, estimated from intertemporal monetary choice questions, does not have the expected (positive) association with obesity.³⁵

Table 12.23 Time discounting and debt holding of obese and non-obese respondents in the 2010 wave data

Conducting an original Internet survey in 2010 (N = 2,351) in which discount rates are estimated from Newton-type sequential binary choice questions, rather than from the reward lists like Table 12.3 in the text, Kang and Ikeda (2013) show that the respondents’ health-related attributes, including body weight, are associated with time discounting as predicted by our previous research. In particular, they successfully show that obesity was positively related to hyperbolic discounting: hyperbolic discounters have a 3.6 percentage-point higher probability of being obese.

However, the above studies are based on non-incentivized responses to hypothetical questions. Several experimental studies have successfully detected the associations between time discounting and body status. In Chabris et al. (2008) and Richards and Hamilton (2012), individual laboratory-measured discount rates are shown to predict inter-personal variations in BMI and other behavioral indices. In both studies, the subjects’ discount factors are estimated to be the hyperbolic type. Unlike our results reported in the preceding article, however, the effects of the degree of impatience and steepness (or hyperbolic discounting) on body weight are not disentangled.

The original title of our article was “Fat debtors” (Ikeda et al. 2009). The empirical observation that obese people tend to have debts was the motivation behind the article. Similarly, Guthrie and Sokolowsky (2012) explore obesity as credit risk and show that the loan delinquency rate among obese people is 20 percent higher than that for the non-obese.

Regarding the relationship between underweight and time discounting, Steinglass and colleagues find that individuals with anorexia nervosa show less temporal discounting than individuals at a healthy weight (Steinglass et al. 2012). Together with our findings, this suggests that being underweight is associated with excessive self-control, rather than a lack of self-control. This relationship is in contrast to other unhealthy behavior and psychiatric disorders, such as smoking and substance abuse, and consistent with the Guthrie and Sokolowsky (2012) finding that underweight people have the lowest delinquency rate in their sample.