2.1. Measuring sophistication

The literature measuring sophistication about time preferences can be organized into three broad strands. The first strand infers sophistication from individuals’ preferences over commitment devices or restricted choice sets. Ahn et al. (Reference Ahn, Iijima, Le Yaouanq and Sarver2019) provide a behavioural characterization of naïvete based on ex-ante choice over menus and ex-post choice from menus: an individual is sophisticated if she is indifferent ex ante between retaining the option to choose from a menu ex post or committing to her actual distribution of choices from that menu. Other work in this strand treats demand for commitment as a ‘smoking gun’ for sophistication (e.g., Ericson & Laibson, Reference Ericson, Laibson, Bernheim, DellaVigna and Laibson2019; Sadoff & Samek, Reference Sadoff and Samek2019; Sprenger, Reference Sprenger2015). But commitment demand is only an outcome of sophistication, not a direct measure. Moreover, Carrera et al. (Reference Carrera, Royer, Stehr, Sydnor and Taubinsky2022) show that experimentally increasing sophistication can actually reduce demand for commitment, contradicting the assumption that commitment demand cleanly reveals sophistication (see also Westphal, Reference Westphal2024).

The second strand measures sophistication more directly by eliciting beliefs about future behaviour and comparing them to realized behaviour. Augenblick and Rabin (Reference Augenblick and Rabin2019) and Fedyk (Reference Fedyk2024) use incentivized real-effort experiments in which participants decide how much to work at different times under varying wages; sophistication is measured by comparing actual work decisions on future dates with participants’ earlier predictions of those same decisions. Both studies find that participants are naïve about their own present bias: they fail to fully anticipate it, leading them to work less than they had predicted. Fedyk (Reference Fedyk2024) further shows that participants are more sophisticated about others’ present bias than about their own, a phenomenon they call ‘asymmetric naïvete’.

Survey-based approaches also belong to this strand. For example, Cobb-Clark et al. (Reference Cobb-Clark, Dahmann, Kamhöfer and Schildberg-Hörisch2024) classify individuals according to the consistency between their ideal, expected, and actual body weight 1 year later. While intuitive, such proxies can misclassify behaviour. For instance, someone who predicts they will not reach their ideal weight would be labelled sophisticated, even if they are simply unmotivated by the goal. More fundamentally, such survey-based classifications do not yield structural parameters such as $\beta$ and $\hat{\beta}$. Sial et al. (Reference Sial, Sydnor and Taubinsky2023) link forecast errors to memory biases, finding that individuals who over-report past attendance are less aware of their time inconsistency. The present paper belongs to this second strand, using incentivized measures of precise behaviours and predictions thereof, while extending prior work by allowing present bias and sophistication to differ across effort and money domains.

The third strand relies on proxies or unincentivized measures of sophistication, such as cognitive ability or related psychological traits. For example, Zhang and Greiner (Reference Zhang and Greiner2021) link cognitive ability to awareness of present bias.

2.2. Identification of time preference parameters

A distinct but related body of work addresses identification challenges in estimating time-preference parameters from intertemporal choice data. Strack and Taubinsky (Reference Strack and Taubinsky2026) highlight important challenges in ‘revision design’ experiments, where participants make intertemporal trade-offs between two periods, deciding first in advance and then again when the time comes. They show that with only ordinal choice data, the same observed behaviour can be rationalized by multiple assumptions about how information about future preferences or constraints is revealed to the decision-maker. They note that when trade-offs are measured in cardinal terms, such as monetary amounts, identification becomes possible. Our design exploits this insight by eliciting choices over monetary amounts across time, enabling us to more precisely recover present bias and sophistication by leveraging variation in participants’ willingness to pay.

2.3. Domain-sensitive time preference parameters

Previous literature shows that time preferences are domain-sensitive. People tend to be more present-biased over primary rewards such as food, juice, or work, while present bias often appears weaker – or even absent – for monetary rewards. For example, Cheung et al. (Reference Cheung, Tymula and Wang2022) show that present bias differs across money, healthy foods, and unhealthy foods.

Of particular relevance to our paper, this domain-sensitivity extends to the money-effort distinction, with evidence suggesting that present bias differs systematically between the two. For example, Augenblick et al. (Reference Augenblick, Niederle and Sprenger2015) find no present bias over money but significant present bias over effort. In a meta-analysis about studies using the Convex Time Budget method, Imai et al. (Reference Imai, Rutter and Camerer2021) show that studies using monetary rewards often show statistically insignificant present bias on average, while effort-based studies tend to report a smaller and significant present bias parameter. Identifying sophistication in Augenblick and Rabin (Reference Augenblick and Rabin2019) and Fedyk (Reference Fedyk2024) requires assuming equal present bias and sophistication over money and effort, thereby yielding estimates that may fail to capture behaviour in either domain. This is what distinguishes the current paper: our experimental design identifies sophistication and effort, without assuming equality across money and effort domains.

3.1. Sessions

Fig. 1 illustrates the sequence of events across the 3 consecutive days of a session. On Day 1’s consent form, participants were informed that the study involved real-effort tasks over 3 days. Participants then read the instructions and completed a comprehension quiz assessing their understanding of the study’s structure – namely, its 3-day duration, the £5 bonus for completing all days, and the fact that all payments (for tasks and study completion) would be made on Day 3. Before making any decisions, they were required to complete training tasks to familiarize themselves with the task format and its difficulty. The information about study and payment structures was reiterated on Days 2 and 3 as a reminder, then they made all decisions and predictions, and then they completed any effort tasks scheduled for that day.

Fig. 1.

Session timeline: the timeline of events within each of the 3 days

Fig. 1 Long description

The flowchart outlines a session timeline across three days. Day 1 includes: 1. Consent and instructions, 2. Comprehension quiz, 3. Training tasks, 4. Decisions and predictions, 5. Effort allocation. Day 2 consists of: 1. Consent, 2. Reminder of the timing of tasks and payments, 3. Decisions and predictions, 4. Effort allocation. Day 3 involves: 1. Consent, 2. Reminder of the timing of tasks and payments, 3. Decisions, 4. Effort allocation, 5. All payments. Arrows indicate the progression from Day 1 to Day 2 and from Day 2 to Day 3.

3.2. Decisions and predictions

Participants were asked to choose and predict the number of time-dated real-effort tasks that they wanted to complete for different wages on the current day and on future days.

Over the 3 days, participants saw 9 decision screens similar to the one shown in Fig. 2. On each of the 9 screens, participants saw 5 different wages or piece rates (1 $p$, 3 $p$, 5 $p$, 7 $p$, and 9 $p$) and had to choose the number of time-dated jobs, between 3 and 13, that they wanted to complete at that wage and on that date. Six of the 9 screens were choice screens, on which participants made their own decisions, while the remaining 3 were prediction screens, on which they predicted the number of jobs their future selves would choose to complete. This means that participants made $5\times 9=45$ decisions in total.

Fig. 2.

Decision screen for choosing jobs $J \in \{1, 2, \dots, 13\}$ at 5 wages. At each wage $w$, for the selected $J$, the screen dynamically calculated and displayed the number of tasks $J(J+1)/2$ and the total corresponding payment $wJ$. For example, in the figure above, the participant selected $J=3$ at $w=1$, and hence the screen said ‘3 Jobs: 6 tasks for 3 pence total’

Fig. 2 Long description

This is a screenshot of a decision screen where participants choose a pay rate for tasks to be completed the day after tomorrow. The screen displays five selectable options, each with a horizontal slider bar and a circular marker indicating selection. Options are listed as follows: 1 penny per job: 6 Jobs, 21 tasks for 6 pence total. 3 pence per job: 6 Jobs, 21 tasks for 18 pence total. 5 pence per job: 7 Jobs, 28 tasks for 35 pence total. 7 pence per job: 8 Jobs, 36 tasks for 56 pence total. 9 pence per job: 9 Jobs, 45 tasks for 81 pence total. Each option is presented on its own line with a slider bar to adjust the number of jobs and tasks.

Jobs translated into real-effort tasks at a non-linear rate: $J$ jobs required completing $J(J + 1)/2$ real-effort transcription tasks (just tasks, henceforth). For example, as seen on Fig. 2, $J=3$ jobs required completing $3(3+1)/2=6$ tasks, $J=5$ jobs required completing $5(5+1)/2=15$ tasks, and $J=13$ jobs required completing $13(13+1)/2=91$ tasks.

As illustrated by Fig. 3, a (transcription) task consisted of typing a sequence of characters into an empty box. If the characters were not typed correctly, an error message appeared and the participant had to retype the characters again. Our transcription tasks resemble those used in Augenblick and Rabin (Reference Augenblick and Rabin2019).Footnote ³

Fig. 3.

Example of a transcription task. Completing $J$ jobs meant finishing $J(J + 1)/2$ transcription tasks for a total payment of $Jp$

3.2.2. All choices and predictions

Table 1 summarizes all the decisions made by participants by date. $J_{t\tau}(w)$ denotes the decisions made on day $t$ about the number of jobs to complete on day $\tau$ at the wage $w$. On Day 1, participants decided the number of jobs to complete on Days 1, 2, and 3, denoted by $J_{11}(w)$, $J_{12}(w)$, and $J_{13}(w)$, respectively. On Day 2, they decided the number of jobs to complete on Days 2 and 3, i.e., $J_{22}(w)$ and $J_{23}(w)$, respectively. On Day 3, they decided how many jobs to complete on Day 3, denoted by $J_{33}(w)$.

Table 1.

Decisions and predictions by date

Table 1 Long description

The table maps when job-count decisions are made and when predictions about those decisions appear, organized by decision date (rows) and work date (columns). On decision date 1, there is a decision for work date 1, and for work dates 2 and 3 the cells include both the decision and a prediction made on decision date 2. On decision date 2, there is a decision for work date 2, and for work date 3 the cell includes both the decision and a prediction made on decision date 3. On decision date 3, there is a decision for work date 3 only. Blank cells indicate no entry for that decision date and work date pairing. The structure implies decisions are made no later than the work date, while predictions are added later for future work dates. Values depend on the wage rate category, but the table does not provide the numeric job counts.

$J_{t\tau}(w)$ denotes the decision made on date $t$ about the number of jobs to do on date $\tau$. $\widehat{J_{t\tau}}(w)$ denotes the prediction about $J_{t\tau}(w)$. $w\in \{1, 3, 5, 7, 9\}$ denotes the wage rate.

On each day $\tau$, one decision was randomly chosen from the pool of decisions made for that day, and participants had to complete the corresponding number of jobs for the corresponding wage. As an example, on Day 3, one decision was randomly drawn from a pool of 15 decisions: decisions $J_{13}$, $J_{23}$, and $J_{33}$, each of them made for 5 different wages.

After making their decisions, participants completed the tasks corresponding to the randomly selected decision for that day. Only after completing these tasks did they proceed to the prediction stage. This temporal separation ensures that decisions and predictions are meaningfully separated in time.

$J^p_{t\tau}(w)$ denotes the prediction of the decision $J_{t\tau}(w)$. Participants faced 3 prediction screens. On Day 1, they made predictions $J^p_{22}(w)$ and $J^p_{23}(w)$ about $J_{22}(w)$ and $J_{23}(w)$, respectively. On Day 2, they predicted $J_{33}(w)$.

3.3. Payments

All payments were made within the evening of the third day after the survey was over, as promised in the instructions. To make sure that participants understood all the sources of payment, we quizzed them about the experiment’s payment structure, including one question about the timing of the payments.Footnote ⁶

There were four sources of payment for each participant. Firstly, participants received a participation fee for each day they participated in the study ( $\unicode{x00A3}2$, $\unicode{x00A3}1$, and $\unicode{x00A3}1$, respectively). The participation fee was higher on the first day than on the other 2 days as participants spent more time reading instructions, completing mandatory tasks for training, and making more decisions. Secondly, participants received a bonus of $\unicode{x00A3}5$ if they completed all 3 days of the experiment. This completion fee was relatively high to minimize attrition. Thirdly, on each day, participants completed tasks and were paid according to one randomly chosen decision out of the decisions they made for that day. To calculate this payment, the randomly drawn wage rate was multiplied by the number of jobs chosen and completed for that decision. On each day, the payment could range from 3 $p$ (1 $p$ $\times$ 3 jobs) to 117 $p$ (9 $p$ $\times$ 13 jobs). On the 3 survey days, participants earned 35 $p$, 35 $p$, and 32 $p$ on average for completing their chosen number of transcription tasks, respectively.

Finally, participants could receive a fourth payment of 20 $p$ if one of their predictions (randomly chosen from their 15 predictions) differed by no more than 2 jobs from their actual decision. Thus, on average, every accurate prediction task only paid 1.33 $p$. Around 87% of all predictions were accurate within 2 jobs, and hence qualified for the 20 $p$ reward if chosen for payment. Unlike Augenblick and Rabin (Reference Augenblick and Rabin2019), who reminded participants of their prior predictions before final decisions, we follow Fedyk (Reference Fedyk2024) in omitting such reminders to avoid priming participants to behave consistently with earlier responses.

Setting the prediction reward involved a trade-off. Too large an accuracy bonus would allow sufficiently sophisticated participants to use the prediction tasks as a commitment device, since the prospect of matching a previously made prediction could incentivize their future selves to work harder than wage incentives alone would warrant. Too small a bonus, on the other hand, might lead participants to provide careless predictions.Footnote ⁷ We therefore set a very small accuracy bonus as a compromise between these two concerns.

3.4. Participants and selection

All experimental sessions were run online in November 2021 and May 2022. With the goal of having a sample of around 100 participants post-attrition, we recruited 138 participants through Prolific. A total of 115 participants completed the study, although 7 had some missing data. In November, the median length of the experiment was 30 minutes on the first day, 15 minutes on the second day, and 9 minutes on the third day. In May, the median length was 27, 12, and 12 minutes, respectively. A total of 108 participants completed all 3 days of the experiment. This study was not preregistered, as the data were collected prior to the widespread adoption of preregistration practices in the field.

Out of the 138 participants who completed the first day of the study, 108 responded to all questions during the 3 days. Of these, 7 participants did not show any variation across any of the 9 screens, leaving 101 participants for meaningful regression analyses. To winsorize this sample, we further excluded the participants who ranked in the top 10% of ‘most frequent violators’ in each of the following categories: wage monotonicity, time-monotonicity with respect to effort date, and time-monotonicity with respect to pay date. Our final dataset therefore consists of 80 participants (out of the 101 regression-eligible). For consistency, and if not otherwise stated, all the tables or graphs that follow use this sample of 80 participants.

The three monotonicity conditions used to define the sample restrictions – wage monotonicity, time-monotonicity with respect to effort date, and time-monotonicity with respect to pay date – are discussed in detail, along with the corresponding statistics, in Online Appendix A.

4.1. Time-consistent vs. sophisticated vs. naïve

Following O’Donoghue and Rabin (Reference O’Donoghue and Rabin1999), we distinguish between three mutually exclusive types of individuals. When $\beta_m = \beta_e = 1$, the individual has exponential discounting preferences. In this case, her relative valuation of earlier versus later utility remains constant regardless of when she is asked to make that trade-off. Since there is no conflict between her preferences at different points in time, such a person is said to be time-consistent, and her plans/predictions automatically align with her future actions.

When either $\beta_m \lt 1$ or $\beta_e \lt 1$, the individual exhibits present bias. When considering trade-offs of utility between two moments, present-biased individuals give stronger relative weight to the earlier moment if it is also in the present. This bias creates the possibility of a divergence between plans/predictions and future actions, depending on whether the individual anticipates her present bias.

O’Donoghue and Rabin (Reference O’Donoghue and Rabin1999) classify individuals as sophisticated or naïve only when they are present-biased (i.e., $\beta_m \lt 1$ or $\beta_e \lt 1$). If the individual correctly anticipates her present bias, she is sophisticated; otherwise, she is naïve.

More generally, we interpret sophistication as the accuracy of individuals’ beliefs about their own present-bias parameters, $\widehat{\beta}_e$ and $\widehat{\beta}_m$, which can be defined irrespective of whether $\beta \lt 1$ or $\beta \geq 1$. A fully sophisticated individual satisfies $\widehat{\beta}_e = \beta_e$ in the effort domain (or $\widehat{\beta}_m = \beta_m$ in the money domain), while deviations between $\widehat{\beta}$ and $\beta$ reflect the degree of naïveté. For completeness, our graphs and tables report both present bias ( $\beta$) and perceived present bias ( $\hat{\beta}$) for all participants, including those with $\beta \geq 1$.

4.2. Identification

Overall, as long as $c(\cdot)$ is convex in $J$, the problem in Equation ((1)) or ((2)) is strictly concave and has a unique solution. In particular, for our parametric estimation exercise, we will assume that $c(x)= \frac{x^{\gamma+1}}{\gamma+1}$ and substitute it in the first-order condition derived from (2):

(3)\begin{equation} \left(\psi \frac{\widehat{\beta}_m^{\mathbb{1} (3 \gt t_d) } w \delta^{3 - t_e}}{\widehat{\beta}_e^{\mathbb{1} (t_e \gt t_d) } } \right) - \frac{2\hat J+1}{2} c'\left(\hat{J}(\hat{J}+1)/2 \right) = 0. \end{equation}

(4)\begin{equation} \iff \left(\psi \frac{\widehat{\beta}_m^{\mathbb{1} (3 \gt t_d) } w \delta^{3 - t_e}}{\widehat{\beta}_e^{\mathbb{1} (t_e \gt t_d) } } \right) - \left( \frac{\hat J(\hat J+1)}{2} \right)^{\gamma} \left( \frac{2\hat J+1}{2} \right) = 0. \end{equation}

Thus, the prediction $\hat{J}$ directly reflects the individual’s beliefs about her own future preferences, as captured by $\widehat{\beta}_m$ and $\widehat{\beta}_e$. This identification strategy relies on other components – such as the marginal cost function $c'(\cdot)$ – being known. This dependency underscores the importance of the training tasks administered before any decisions were made on Day 1 (see Fig. 1), as they were designed to increase participants’ familiarity with both the overall cost and the marginal cost of performing the tasks. As before, one can move between the optimality conditions for $J^\ast$ and $\hat J$ by interchanging $\beta_m$ with $\widehat{\beta}_m$, and $\beta_e$ with $\widehat{\beta}_e$.

4.3. Our approach

To jointly estimate $\beta$ and $\widehat {\beta}$ parameters separately for effort and money, our experimental design allows us to independently observe:

• • Variation 1: Choices and predictions when the effort costs are in the future (tasks to be completed on later days) versus when the effort costs are immediate (tasks to be completed the same day).

• • Variation 2: Choices and predictions when the benefits (payments) are in the future (decisions made on Days 1 and 2, paid on Day 3) versus when the benefits are immediate (decisions made on Day 3, paid the same day).

While other studies such as Augenblick and Rabin (Reference Augenblick and Rabin2019) and Fedyk (Reference Fedyk2024) also use Variation 1, they lack Variation 2, as their payment date is always strictly after every effort date. Thus, $\beta_m$ and $\widehat{\beta}_m$ can never be identified separately from $\beta_e$ and $\widehat{\beta}_e$, respectively.

Equipped with Variations 1 and 2, and assuming i.i.d. normally distributed errors with equal variance across choice and predictions, we estimate the parameter vector of the model, i.e., $\theta = (\beta_e, \widehat{\beta}_e, \beta_m, \widehat{\beta}_m, \delta, \psi, \gamma)$, using the maximum likelihood estimator (MLE) over both choice and prediction data. $\widehat{\theta}_{\text{MLE}}$, the maximum likelihood estimate of $\theta$, is defined as

\begin{align*} \widehat{\theta}_{\text{MLE}} & = \arg\max_{\theta} \sum_{f=1}^F \ln \mathcal{L}_f (\theta), \quad \text{where} \nonumber \\ \ln \mathcal{L}_f (\theta) & = \begin{cases} - \ln \sigma - 0.5 \ln 2\pi - 0.5 \left( \frac{J_f^{\ast}(\theta) - J_f}{\sigma} \right)^2 &\text{For choice data}\\ - \ln \sigma - 0.5 \ln 2\pi - 0.5 \left( \frac{\widehat{J_f(\theta)} - J_f}{\sigma} \right)^2 &\text{For prediction data} \end{cases} \end{align*}

where $f$ is an index for choice and prediction data points, combined across all individuals and 3 days for aggregate analysis, and fixing an individual over 3 days for individual-level analysis. For any value of $\theta$, $J_f^{\ast}(\theta), \widehat{J_f(\theta)}$ are calculated using Equation (4). MLE finds the set of parameters $\theta$ that maximizes the joint likelihood of observing the complete data.

5.2.1. Visualizing the structural identification

Before presenting the results from our structural estimation, we plot prominent features of the aggregate data that provide an intuition about how the parameters of the structural model are identified. For simplicity, we assume $\delta=1$.Footnote ¹¹ Throughout this section, we measure effort in tasks rather than jobs, as the empirical CDFs are smoother and easier to read under this scale given the non-linear mapping $e = J(J+1)/2$; all results are qualitatively identical when effort is measured in jobs instead.

Let $e_{t\tau}$ be the effort chosen (measured either in jobs or in tasks) on day $t$ for day $\tau$. For a fixed decision period $s$, the comparison between choices for immediate effort ( $e_{ss}$) and choices for future effort ( $e_{st}$ for $t \gt s$), both made at time $s$, identifies $\beta_e$. Under $\beta_e \leq 1$, we would observe $e_{st} \gt e_{ss}$ for all $t \gt s$, and under $\beta_e = 1$, we would observe $e_{st}=e_{ss}$. In the left panel of Fig. 7, we compare the empirical CDFs of immediate effort $e_{ss}$, as measured in tasks, and future effort $e_{st}$ while pooling over wage rates and decision days ( $ss=11,22,33$ and $st=12,23$).Footnote ¹² Visually, $e_{ss}$ is first-order stochastically dominated by $e_{st}$, which suggests that fewer tasks are consistently allocated to the immediate day. Based on these patterns, ex-ante, we expect to estimate $\beta_e$ to be significantly below 1.

Fig. 7.

The left figure plots immediate versus future effort and helps identify $\beta_e$. The sample size is 1305 for both future and immediate effort. The right figure plots $e_{22}$, $\widehat{e}_{23}$, $e_{33}$, $\widehat{e}_{22}$, and helps identify $\widehat{\beta}_e, \beta_m, \widehat{\beta}_m$. The sample size is 435 for all four variables

Fig. 7 Long description

The image contains two step line graphs side by side. The left graph compares Immediate and Future task allocation. The vertical axis is labeled Cumulative Probability, ranging from 0 to 1. The horizontal axis is labeled Tasks, ranging from 0 to 100. The Immediate line is solid and the Future line is dashed. Immediate is higher than Future at several points, notably around 20 and 50 tasks, converging near 90 tasks. The right graph compares e subscript 22, e subscript 33, e hat subscript 23 and e hat subscript 22. The vertical axis is Cumulative Probability, ranging from 0 to 1 and the horizontal axis is Tasks, ranging from 0 to 100. The lines are distinguished by styles: dashed for e subscript 22, solid for e subscript 33, dotted for e hat subscript 23 and solid for e hat subscript 22. All lines show similar trends, with notable differences around 30 and 70 tasks. The purpose is to illustrate differences in task allocation strategies and their cumulative probabilities over a range of tasks.

Next, we explain which data patterns reveal three key parameters: $\widehat{\beta}_e, \beta_m, \widehat{\beta}_m$. Sophistication over effort, $\widehat{\beta}_e \in [\beta_e,1)$, is revealed if the participants predict the difference between immediate and future choices. In particular, under $\widehat{\beta}_e \leq 1$, we would observe $\widehat{e_{22}} \lt \widehat{e_{23}}$, where $\widehat{e_{22}}$ is the effort that on Day 1 participants predict they will choose on Day 2 to be completed on Day 2, while $\widehat{e_{23}}$ is the effort that on Day 1 participants predict they will choose on Day 2 to be completed on Day 3. Under $\widehat{\beta}_e = 1$ we would observe $\widehat{e_{22}}=\widehat{e_{23}}$. Hence, comparing $\widehat{e_{22}}, \widehat{e_{23}}$ reveals $\widehat{\beta}_e$. The present bias for money, $\beta_m$, is revealed by comparing two immediate choices, $e_{33}$ and $e_{22}$, where $e_{33}$ pays on the same day and hence is not affected by $\beta_m$. Under $\beta_m \lt 1$, we would observe $e_{33} \gt e_{22}$ and under $\beta_m=1$ we would observe $e_{33}=e_{22}$. Finally, the prediction of immediate effort $\widehat{e_{22}}$ is not affected by $\beta_e$ but is affected by $\widehat{\beta}_m$. Thus, $\widehat{\beta}_m$ is revealed by the comparison between $e_{33}$ and $\widehat{e_{22}}$. Under $\widehat{\beta}_m \lt 1$, we would observe $e_{33} \gt \widehat{e_{22}}$ and under $\widehat{\beta}_m=1$ we would observe $e_{33}=\widehat{e_{22}}$.

To provide intuition about $\widehat{\beta}_e,\ \beta_m,\ \widehat{\beta}_m$, we plot the empirical CDFs of $e_{22},\ \widehat{e_{23}},\ e_{33},\ \widehat{e_{22}}$ in the right panel of Fig. 7. The distributions are all close to each other and intertwined, with none first-order stochastically dominating the others. None of the three pairwise comparisons that identify $\widehat{\beta}_e, \beta_m, \widehat{\beta}_m \lt 1$, namely $\widehat{e_{22}}\ vs\ \widehat{e_{23}}$, $e_{33}\ vs\ e_{22}$, $e_{33}\ vs\ \widehat{e_{22}}$, are significantly different. These patterns suggest that, ex-ante, we should not expect to estimate $\widehat{\beta}_e, \beta_m, \widehat{\beta}_m$ values that are different from 1.

5.2.2. Aggregate analysis

Table 3 presents the structural estimates at the aggregate level, using the MLE described in Section 4. For each parameter, we report the 95% confidence interval instead of the standard error, to make it easy to understand if the coefficients are significantly different from 1 or not.

Table 3.

Structural estimates at the aggregate level

Table 3 Long description

The table reports aggregate structural parameter estimates under four model variants, with point estimates and 95 percent confidence intervals for most parameters. In the model that constrains effort and money to share the same present bias, present bias is 0.71 and sophistication is 0.74. When effort and money are estimated separately, present bias is about 0.70 for effort but about 0.95 for money, and sophistication is about 0.80 for effort and about 0.95 for money. In the version without prediction, effort present bias remains about 0.71 and money present bias about 0.93, but sophistication is not estimated. In the variant that fixes the long run discounting to one, effort present bias rises to 0.79 and money present bias to 1.09, with sophistication about 0.88 for effort and 1.04 for money. The convexity parameter is stable across all models at roughly 2.56 to 2.58, the effort scale is around 199 to 212 with wide intervals, and the error size is about 3.22 to 3.23 with tight intervals. Log likelihood values are similar across the three models using 3,600 observations, while the no prediction model uses 2,400 observations; all models use 80 participants. Confidence intervals indicate substantial uncertainty for some parameters, especially the effort scale and money-domain present bias in the separate model.

[1] Same follows Augenblick and Rabin (Reference Augenblick and Rabin2019) to assume that present bias or sophistication over effort and money are identical. [2] Separate generalizes Same by allowing for different present bias and sophistication parameters across effort and money domains. [3] No prediction is similar to Separate, but it excludes predictions and hence cannot estimate sophistication. [4] Separate, $\delta=1$ imposes that the long-run discount factor is one. The numbers in square brackets indicate the lower and upper bounds of the 95% confidence interval.

The first column (‘Same’) follows the assumption from Augenblick and Rabin (Reference Augenblick and Rabin2019) that present bias and sophistication are identical across money and effort. Thus, this specification replicates their identification assumption and serves as a counterfactual. The estimated present bias parameter is $0.71$ and the 95% confidence interval lies below $1$. The sophistication parameter is $0.74$ and the 95% confidence interval lies below 1 as well.

The second column (‘Separate’) is our novel specification: it allows for different present bias and sophistication over effort and money. We find that $\beta_e$ is $0.70$ and significantly different from 1. Interestingly, the common $\beta$ estimated in the ‘Same’ specification mimics the $\beta_e$ in the ‘Separate’ specification. $\widehat{\beta}_e$ is $0.80$, which is higher than $\beta_e=0.70$ (although the difference is not statistically significant) and indicates partial sophistication over effort. Neither of the money-domain parameters $\widehat{\beta}_m={\beta}_m=0.95$ are estimated to be significantly different from 1.

Researchers interested in estimating only the present bias parameters could use a variant of our design that does not include the prediction tasks. To approximate the results of a design with no predictions, in the third column (‘No prediction’), we run our analysis while excluding the prediction tasks (i.e., using only the data from the 6 decision screens that are not predictions). The results are very similar to the results from the ‘Separate’ specification. This is reassuring, as it indicates that including the prediction tasks does not bias the estimates of present bias.

The 95% confidence interval of $\delta$ contains 1 consistently across all three specifications. Motivated by this finding, in the fourth column (‘Separate, $\delta=1$’), we re-estimate the ‘Separate’ specification assuming $\delta$ to be equal to 1 and including both choice and prediction data. While this increases the estimates of present bias over both domains, $\beta_e$ is still estimated to be significantly below $1$. For completeness, we note that imposing $\delta = 1$ on the restricted specification of column [1] yields a common $\hat{\beta}$ closer to 1, consistent with the pattern observed when comparing columns [2] and [4].Footnote ¹³

Model comparison by Likelihood Ratio (LR) test and Akaike Information Criterion (AIC)

Under the null hypothesis that the simpler nested model in specification [1] (‘Same’) fits the data as well as the model in specification [2] (‘Separate’), the LR test statistic is 10.92, which should be distributed as chi-squared with 2 degrees of freedom. We obtain a corresponding p-value $ \lt 0.01$ and thus we have statistically significant evidence to reject the null and conclude that specification [2] fits the data better than specification [1]. On the other hand, the LR test statistic in the comparison of specifications [2] and [4] is 1, which should be distributed as chi-squared with 1 degree of freedom. The corresponding p-value is $0.68$ and we do not find significant evidence for model [2] fitting the aggregate data better than the simpler nested model [4]. Thus [4] fits the data almost as well as any other model without overfitting the data. Akaike Information Criterion (AIC) for model selection also suggests [4] as the superior model among [1], [2], and [4].

Stability of estimates

All estimated parameters are relatively stable across all specifications. For example, under all specifications, $\beta_e$ is significantly below 1, and $\widehat{\beta}_e$ and $\beta_m$ are no different from 1. Similarly, the convexity parameter $\gamma$ is consistently above 1, implying that the cost function is significantly convex.Footnote ¹⁴ The scale of effort $\psi$ is also similar across all specifications, implying that the baseline level of effort did not vary with the type of model chosen.

Conflated estimates from model [1]

In model [1], MLE tries to fit the misspecified model to the data by biasing $\delta$ upwards (increasing $\delta$ significantly over 1). The common $\beta$ is closer to $\beta_e$ than to $\beta_m$. Further, the estimated $\beta$ and $\widehat{\beta}$ are almost identical in magnitude under [1], implying a high degree of estimated relative sophistication. In fact, [1] is the only specification that rejects complete naïvete ( $\widehat{\beta}=1$) in favour of $\widehat{\beta} \lt 1$. As a result, participants are estimated as relatively more sophisticated under [1] than in all other specifications. For further details about how the misspecified model might generate biased estimates, see Online Appendix C.2.

5.2.3. Individual analysis

Next, we estimate the time preference parameters for each individual separately. Fig. 8 plots the distributions of the present bias and sophistication parameters, in both effort and money domains, for 77 participants.Footnote ¹⁵ For present bias over effort, the distribution of $\beta_{e}$ is skewed toward values below 1, with over 67% of participants exhibiting present bias in the effort domain, a median of 0.91, and a mean of 0.98. For four participants, $\beta_{e}$ is less than 0.1.Footnote ¹⁶ The distribution of sophistication over effort, $\hat{\beta}_{e}$, is similarly concentrated below 1 but less dispersed, with a median of 0.94 and a mean of 1.07, consistent with partial sophistication at the individual level. By contrast, the distributions of both $\beta_{m}$ and $\hat{\beta}_{m}$ are more symmetric and centered closer to 1, with medians of 1.04 and 1.05 respectively, confirming that present bias and sophistication over money are on average weaker than in the effort domain. Still, there is substantial heterogeneity, with some participants exhibiting present bias in the money domain as well.

Fig. 8.

Structural estimates of present bias and perceived present bias over effort and money. The sample size is 77 for all four graphs

Fig. 8 Long description

The image A showing a histogram labeled Actual Present Bias (effort). The vertical axis label is Density, ranging from 0 to 1. The horizontal axis ranges from 0 to 8. The bars are grouped into narrow adjacent bins along the horizontal axis. The tallest bars are between 0 and 1.5, with the highest bar around 1. The bars decrease after about 1.5, with small bars between about 2 and 3. There is a single small bar around 5. The image B showing a histogram labeled Actual Present Bias (money). The vertical axis label is Density, ranging from 0 to 0.8. The horizontal axis ranges from 0 to 8. The bars are grouped into narrow adjacent bins along the horizontal axis. The tallest bars are between 0 and 1.5, with the highest bar around 1. Bars continue at lower heights between about 1.5 and 3. There are isolated small bars around 4, 5, 6, 7 and 8. The image C showing a histogram labeled Perceived Present Bias (effort). The vertical axis label is Density, ranging from 0 to 1. The horizontal axis ranges from 0 to 8. The bars are grouped into narrow adjacent bins along the horizontal axis. The tallest bars are between 0 and 1.5, with the highest bar around 1. Bars decrease after about 1.5, with small bars between about 2 and 3. There are small isolated bars around 3.5 to 4.5. The image D showing a histogram labeled Perceived Present Bias (money). The vertical axis label is Density, ranging from 0 to 0.8. The horizontal axis ranges from 0 to 8. The bars are grouped into narrow adjacent bins along the horizontal axis. The tallest bars are between 0 and 1.5, with the highest bar around 1. Bars continue at lower heights between about 1.5 and 3. There are isolated small bars around 4 to 6.

Fig. 9 plots the cumulative distribution functions of the individual-level structural estimates. In the effort domain (top-left panel), the distribution of present bias ( $\beta_{e}$) lies to the left of the corresponding sophistication measure ( $\widehat{\beta}_e$), indicating that participants exhibit present bias but only partially anticipate it. In the money domain (top-right panel), the distributions of $\beta_{m}$ and $\widehat{\beta}_m$ largely overlap, suggesting no systematic gap between actual and perceived bias. The bottom panels compare estimates across domains: present bias is somewhat stronger for effort than for money (bottom-left), as the effort distribution is shifted toward lower values. In contrast, the sophistication distributions (bottom-right) largely overlap, providing little evidence of systematic differences across domains.

Fig. 9.

Comparison of cumulative density functions of structural estimates of present bias and sophistication over effort and money, estimated at the individual level. The sample size is 77 for all four graphs

Fig. 9 Long description

The image A showing a line graph. Vertical axis label: Cumulative probability. Vertical axis range: 0 to 1 with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Horizontal axis tick labels: 0, 1, 2, 3, 4, 5. Horizontal axis variable name and unit are not shown. Legend text: Actual Present Bias (effort); Perceived Present Bias (effort). Two curves are shown, one solid and one dashed. Both curves rise from near 0 at horizontal axis value 0 to near 1 by about horizontal axis value 3 to 5. The solid curve is higher than the dashed curve for much of the rise between about 0.5 and 2 on the horizontal axis. Around cumulative probability 0.8, both curves are near horizontal axis values around 1.5 to 2. The image B showing a line graph. Vertical axis label: Cumulative probability. Vertical axis range: 0 to 1 with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Horizontal axis tick labels: 0, 2, 4, 6, 8. Horizontal axis variable name and unit are not shown. Legend text: Actual Present Bias (money); Perceived Present Bias (money). Two curves are shown, one solid and one dashed. Both curves rise steeply between about horizontal axis values 0.5 and 2, then flatten near cumulative probability 0.9 to 1 from about horizontal axis value 3 to 8. The dashed curve is slightly lower than the solid curve during the steep rise. Around cumulative probability 0.8, both curves are near horizontal axis values around 1.5 to 2. The image C showing a line graph. Vertical axis label: Cumulative probability. Vertical axis range: 0 to 1 with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Horizontal axis tick labels: 0, 2, 4, 6, 8. Horizontal axis variable name and unit are not shown. Legend text: Actual Present Bias (effort); Actual Present Bias (money). Two solid curves are shown. Both curves rise from near 0 at horizontal axis value 0 to near 1 by about horizontal axis value 4 to 8. The effort curve is higher than the money curve between about horizontal axis values 0.5 and 2. Around cumulative probability 0.8, the effort curve is near horizontal axis value around 1.5 and the money curve is near horizontal axis value around 2. The image D showing a line graph. Vertical axis label: Cumulative probability. Vertical axis range: 0 to 1 with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Horizontal axis tick labels: 0, 1, 2, 3, 4, 5. Horizontal axis variable name and unit are not shown. Legend text: Perceived Present Bias (effort); Perceived Present Bias (money). Two dashed curves are shown. Both curves rise from near 0 at horizontal axis value 0 to near 1 by about horizontal axis value 3 to 5. The effort curve is higher than the money curve between about horizontal axis values 0.5 and 2. Around cumulative probability 0.8, the effort curve is near horizontal axis value around 1.5 to 2 and the money curve is near horizontal axis value around 2. Across the four graphs, each plot uses the same vertical axis label Cumulative probability with range 0 to 1 and each plot compares two curves using solid versus dashed line styles or two solid lines. The effort versus money comparisons appear in the graphs with legends that pair effort and money series.

Fig. 10 presents four scatterplots of the estimated individual-level present bias and sophistication parameters in both domains. The top-left panel presents the scatter plot of $\beta_e$ and $\widehat{\beta}_e$. The correlation between the two parameters is high (Spearman coefficient = 0.77, $p \lt .0001$) and much higher than the range 0.24–0.28 reported in Augenblick and Rabin (Reference Augenblick and Rabin2019) as the correlation coefficient between $\beta$ and $\widehat{\beta}$. The correlation suggests that participants are at least partially aware of their own present bias over effort.Footnote ¹⁷ The discrepancy between our results and those of Augenblick and Rabin (Reference Augenblick and Rabin2019) may be partly explained by the fact that their estimate of correlation also depends on the correlation between $\beta_m$ and $\widehat{\beta}_m$. The top-right panel presents the scatter plot between $\beta_m$ and $\widehat{\beta}_m$. The correlation between these parameters is 0.38 ( $p=0.0006$).Footnote ¹⁸ The bottom-left panel shows a strong and positive relationship between $\beta_e$ and $\beta_m$ (Spearman coefficient=0.77, $p \lt 0.0001$). Finally, the bottom-right panel presents the scatter plot of $\widehat{\beta}_e$ and $\widehat{\beta}_m$. The correlation is 0.30 ( $p=0.0068$).

The low correlation in the bottom-right panel is informative: knowing how sophisticated a participant is in the effort domain conveys little information about how sophisticated they are in the money domain. This contrasts with the strong correlation between $\beta_e$ and $\beta_m$, indicating that underlying biases are correlated across domains even when self-awareness about those biases is not.

Further, the parametric estimates of $\beta_e$ and $\widehat{\beta}_e$ are strongly correlated with the reduced-form estimates of present bias and perceived present bias over effort, respectively, obtained in Section 5.1.2. Both correlations are 0.94 ( $p \lt 0.0001$). This provides in-sample validation of the structural parameter estimates.

In the effort domain (top-left panel of Fig. 10), 39 subjects both revealed and predicted present bias ( $\beta_e \lt 1$, $\widehat{\beta}_e \lt 1$) while 20 both revealed and predicted future bias ( $\beta_e \gt 1$, $\widehat{\beta}_e \gt 1$). Thus, 59 of 77 participants correctly anticipated the direction of their bias. In the money domain, the corresponding number is 57 of 77. Table 9 in the Online Appendix further confirms this pattern by reporting the distribution of sophistication estimates conditional on whether participants were present-biased or future-biased. In both domains, participants’ predictions track the sign of their bias: those with $\beta \lt 1$ predict $\widehat{\beta} \lt 1$, while those with $\beta \geq 1$ predict $\widehat{\beta} \gt 1$. This pattern – present across roughly three quarters of all participants – points to partial sophistication in a broader sense: participants anticipate the direction of their bias, but not its full magnitude.

Fig. 10.

Comparison of structural estimates of present bias and sophistication across domains. The points above and below the 45-degree line are marked with a circle and check mark respectively. The sample size is 77 for all four graphs. All panels indicate the number of participants with parameter values less than 1 or greater than 1 along the respective axes

Fig. 10 Long description

The image contains four scatter plots. Each plot is divided into four quadrants by dashed reference lines drawn at the value 1 on both the horizontal and vertical axes. Two types of data points are plotted across all four graphs: open circles and cross marks. Each quadrant displays a count labeled as N, indicating the number of participants with values either less than or greater than 1 along each axis. Plot 1 compares actual present bias (effort) on the horizontal axis against perceived present bias (effort) on the vertical axis. Both axes extend from 0 to 6. The quadrant counts are: lower-left N equals 46, upper-left N equals 14, upper-right N equals 21 and lower-right N equals 6. The largest concentration of points falls in the lower-left quadrant, near the origin, where both actual and perceived values are below 1. Several points extend beyond 1 on both axes into the upper-right quadrant. Notable outliers appear near values of 4 to 5 on the horizontal axis and values of 3.5 to 4 on the vertical axis. Plot 2 compares actual present bias (money) on the horizontal axis against perceived present bias (money) on the vertical axis. Both axes extend from 0 to approximately 7. The quadrant counts are: lower-left N equals 28, upper-left N equals 16, upper-right N equals 34 and lower-right N equals 9. A dense cluster of points appears near the origin, with a notable spread into the upper-right quadrant. Outliers are visible near horizontal axis values of 6 to 7 and vertical axis values of 4 to 5. Plot 3 compares actual present bias (effort) on the horizontal axis against actual present bias (money) on the vertical axis. Both axes extend from 0 to 6. The quadrant counts are: lower-left N equals 40, upper-left N equals 20, upper-right N equals 23 and lower-right N equals 4. Points are heavily concentrated in the lower-left quadrant. Outliers appear near a vertical axis value of 7 and a horizontal axis value of 5. Plot 4 compares perceived present bias (effort) on the horizontal axis against perceived present bias (money) on the vertical axis. Both axes extend from 0 to 6. The quadrant counts are: lower-left N equals 28, upper-left N equals 24, upper-right N equals 26 and lower-right N equals 9. Points are distributed across the lower-left and upper-left quadrants, with a spread into the upper-right quadrant. Outliers appear near vertical axis values of 5 to 6 and horizontal axis values of 3 to 4.