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Holt & Laury’s (H&L) Multiple Price List Design
Of the 55 tasks designed to measure risk aversion, the first 30 are of the Holt and Laury (H&L) type invented by Miller et al. (1969) and used in Holt and Laury (2002). Choice payments and probabilities are presented using an inuitive pie chart representation popularized by Hey and Orme (1994). There are 3 groups of 10 questions. In each group of questions, subjects are pre-sented with an ordered array of binary lottery choices. In each choice task they choose between lottery A (safer) and lottery B (riskier). In each subsequent row, the probability of the higher payoff in both lotteries increases in increments of 0.1. While the expected value of both lotteries increases, the riskier option becomes relatively more attractive. As in the first row of each set of questions the expected value of the safer lottery A is greater than that of the riskier lottery B, all but risk-seeking individuals should choose the safer option. Midway through the 10 questions, the expected value of the riskier lottery B becomes greater than that of the safer lottery A. At this point, risk neutral subjects should switch from the safer to the riskier option. In the remain-ing rows the relative attractiveness of lottery B steadily increases until it becomes the dominant choice in the last row.6 By the last row of each set of H&L questions, all individuals are expected to have switched to the riskier option. Each person’s “switching point” should be indicative of his risk aversion. By design, in the absence of a shock to either his preferences or utility, each indi-vidual should switch at exactly the same point of the 3 sets of H&L questions. The fact that this is not the case in reality highlights the need to use a model which allows for some randomness in decision-making.
Binswanger’s Ordered Lottery Selection (OLS) design
The remaining 25 tasks designed to measure risk aversion used in this study are of the ordered lottery selection (OLS) design developed by Binswanger (1980) and used by Eckel and Grossman (2002 and 2008). They consist of 5 groups of 5 questions. Once again, in each group of questions, subjects are presented with an ordered array of binary lottery choices. In each choice task they choose between lottery A (safer) and lottery B (riskier). This time, lottery A offers a certain amount in the first row and all other alternatives increase in expected payoff but also in its variance. In each subsequent row the riskier option becomes relatively less attractive. Individuals are thus expected to switch from the risky to the safe option at some point (assuming that they initially picked the risky option). Once more, the “switching point” should be indicative of each individual’s risk preferences. By design, the switching point for a given individual should vary among the 5 sets of OLS type questions, unlike in the H&L type ones. In the absence of stochastic shocks to utilities of preferences, the H&L tasks should allow for the identification of an interval for an individual’s risk aversion while the OLS tasks should permit the refinement of this interval. Furthermore, while the H&L tasks focus on the most common range of risk preferences (up to a coefficient of risk aversion of 1.37), MPL tasks let us identify highly risk-averse individuals.
Harisson and Rutstrom (2008) find a risk-aversion of 0.75 using H&L type tasks and of 0.66 using OLS type tasks for the same sample of individuals. However, the estimate is less precise using OLS type questions. They thus conclude that “[t]he results indicate consistency in the elicitation of risk attitudes, at least at the level of the inferred sample distribution”. Both types of lottery choice tasks are thus treated the same in the structural model.
Temporal Choice Tasks
All 48 questions designed to elicit time preferences are of the type used in Coller and Williams (1999). They consist of 8 groups of 6 questions. In each group of questions, subjects are presented with an ordered array of binary choices. In each choice task they choose between an immediate payment7 and a future payment. In each subsequent row the magnitude of the future payment increases. Most individuals are thus expected to switch from the immediate to the future payment at some point. The “switching point” should be indicative of each individual’s time preferences. By design and in the absence of stochastic shocks, each individual should have one switching point in the first 4 sets of temporal choice tasks and another one in the 2nd set. If information on his risk aversion is available, the two sets of tasks designed to elicit time preference should thus yield two (overlapping) intervals for his discount rate.
Observed Individual Choices
Figure 1 plots the distributions of individuals’ choices on tasks designed to elicit their preferences. It shows that that there is significant heterogeneity in choices and that extremes of both distribu-tions (all risky or all safe and all immediate or all distant payments) have non-zero mass. While on the lottery choice tasks the distribution roughly resembles normality this is not the case on temporal choice tasks. The latter distribution is very wide and has high mass points at the ex-tremes. Around 10% of the overall population choose either all immediate payments or all distant payments. Particularly striking is the large share of seemingly very impatient people. However, as mentioned before, one needs to have estimates of individuals’ risk aversion in order to be able to draw clear conclusions about their discount rates.
Background Information
The experiment also solicits a large amount of background information both from students and from their parents. The collected information includes grades, a measure of intelligence, measures of non-verbal ability, personality, finances, school and job aspirations, etc. See Section 9.a of the Appendix for a list of measures selected to approximate cognitive ability and 3 of the Big Five personality traits and of the loadings associated with each measure of these factors. The magnitudes of the loadings vary widely. This shows that some indicators are better measures of the underlying ability and personality traits than others. It confirms the usefulness of using a factor model to address measurement errors inherent in measures of ability and personality (see for example Cunha and Heckman, 2009).For more information on the experiment, see Belzil et al. (2016) or Johnson and Montmarquette (2015).
Model
Before providing technical details, let us expose the general set-up of the model. As described in the previous section, every individual i performs a large number of choice tasks. Each choice task consists of a binary choice. In some cases, the choice is made between lotteries with different expected payoffs and variances and therefore provides information about an individual’s specific risk aversion parameter. In other cases, the choice is between an early (immediate) payment and a later payment. In conjunction with the risk aversion estimate, it can be used to identify an individual’s discount rate. The lottery choice tasks are indexed by l and the temporal choice tasks are indexed by t. Because individuals perform a large number of tasks, and in line with the Random Preference Model (RPM), I introduce two stochastic shocks (one for each preference parameter) and assume that each preference parameter is hit by one of the possible realizations of these shocks every time a task is performed. These shocks are independent across tasks and across individuals. Formally, this entails assuming that both risk aversion and the discount rate are random variables from whose distributions a particular realization is drawn every time a choice task is performed. This could be due to actual preference instability, imperfect self-knowledge, or measurement error.
Because I have access to a large number of psychometric measurements for the individuals who performed these choice tasks, I can investigate the existence of a mapping from individual-specific preference parameters onto psychological traits using a factor model. Unlike what has been pre-viously done in the literature, I extend the notion of preference heterogeneity to also incorporate heterogeneity in the stability of individual preferences and in seemingly irrational choices. This approach allows one to differentiate between heterogeneity in the curvature of the utility function (or in discount rates) and heterogeneity in parameters capturing stochastic behavior.
Ability and the psychological traits (which I shall refer to as factors) are themselves unobserved. They are, however, noisily measured by observed indicators proper to each individual. This data structure makes it amenable to study using factor analysis. I estimate risk-aversion and time-preference parameters jointly with the factor distributions for maximum efficiency. I then relate all components of the model in a structural framework where preference and rationality param-eters are a function of observed characteristics, underlying factors, and pure unobserved hetero-geneity. The following sections describe in turn each of the building blocks of the model.
Preferences
In the RPM framework, an individual’s preference parameter is hit by a random shock in each choice task he faces. His “instantaneous” preference is thus composed of an average deterministic part and of a random shock †i,t which hits individual i in each task t. This essentially makes the preference parameter a random variable centered around its expected value for each individual.
Utility is assumed to be constant relative risk aversion (CRRA). To simplify, I assume 0 back-ground consumption ! and U(!) ˘ 0 as in Apesteguia and Ballester (2018).9
Risk Aversion
Risk aversion, in its most basic sense, can be defined such that if an individual is faced with two choices one of which is riskier, his probability of picking the riskier option decreases as his risk aversion rises. A convincing model of choice under risk should therefore predict a monotonically decreasing relationship between the probability of choosing the riskier option and aversion to risk. Apesteguia and Ballester (2018) demonstrate that the Random Utility Model (RUM) used almost exclusively in previous literature to estimate risk preferences does not satisfy this condition. The RPM, on the other hand, does.
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