Risk preferences and development revisited

The results presented in this paper are stable to using most major theories, including expected utility theory (EUT) and prospect theory (PT). The main modelling approach in the paper is motivated by obtaining the highest possible descriptive accuracy, conditional on keeping the analysis tractable from an empirical point of view. We will adopt a reference-dependent modelling approach throughout. Such an approach is an integral part of prospect theory (Kahneman and Tversky 1979). It was first proposed for expected utility by Markowitz (1952), in an attempt to accommodate the observation of lottery and insurance purchase by the same person, and has by now been widely adopted into expected utility models in both theory and empirical analysis (Diecidue and van de Ven 2008; Kőszegi and Rabin 2007; Sugden 2003; von Gaudecker et al. 2011).⁴ The elicitation tasks are designed in such a way as to fix the reference point to zero—see L’Haridon and Vieider (2018) for a more detailed discussion.

We will start by representing our preferences through a PT model. We describe decisions for binary prospects. For outcomes that fall purely into one domain, i.e., \(x>y\ge 0\) or \(0\ge y>x\), we can represent the utility of a prospect \(\xi _i\), \(U(\xi _i)\), as follows:whereby the probability weighting function w(p) is a strictly increasing function that maps probabilities into decision weights, and which satisfies \(w(0)=0\) and \(w(1)=1\); the superscript j indicates the decision domain and can take the values \(+\) for gains and − for losses; and v(.) represents a utility or value function which indicates preferences over outcomes, with a fixed point such that \(v(0)=0\), and \(v(x)=-v(-x)\) if \(x<0\). Contrary to expected utility models, utility curvature in the full PT model cannot be automatically equated with risk preferences, since the latter are determined jointly by the utility function and the weighting function (Schmidt and Zank 2008). For mixed prospects, where \(x>0>y\), the utility of the prospect can be represented as:In our experimental tasks, we elicit certainty equivalents, such that by definition \(ce\sim (x,p;y)\), where \(\sim \) indicates indifference. We can represent the certainty equivalents estimated according to the model just presented as follows:To specify the model set out above, we need to determine the functional forms to be used. We start by assuming utility to be piecewise linear:where the parameter \(\lambda \) indicates loss aversion, generally represented as a kink in the utility function at the origin (Abdellaoui et al. 2007; Köbberling and Wakker 2005).

Simplifying our model by assuming utility to be piecewise linear has several advantages in our setup, which we belief more than outweigh potential drawbacks. Yaari (1987) powerfully made the point that representing risk preferences through subjective probability transformations is just as legitimate as representing them through outcome transformations, and may be more psychologically accurate for small stakes. Our model is then the natural extension of Yaari’s dual theory to reference-dependent models (see Schmidt and Zank 2007, for an axiomatization of this model). Most importantly, this assumption allows us to directly compare subject pools in terms of their risk preferences. This is much more difficult using a full prospect theory model, given issues of collinearity between utility and weighting functions, which may both reflect risk preferences under prospect theory (Zeisberger et al. 2012). The obvious cost of our simplification is that we largely ignore any variation taking place over different stake levels. This variation is, however, modest in our data, and most interesting patterns with our stake levels typically emerge over the probability dimension (Fehr-Duda and Epper 2012; Prelec 1998). Most importantly, none of the results presented below depend on the linear probability assumption (see Appendix A for a stability analysis using the full PT model).

For probability weighting, we adopt the 2-parameter weighting function proposed by Prelec (1998). (Other functional forms from the two-parameter family deliver similar results. One-parameter forms are, on the other hand, not well suited to describe our data, for reasons that will become apparent below):For \(\beta =1\), this function conveniently simplifies to the 1-parameter function proposed by Prelec, which has a fixed point at \(1/e\simeq 0.368\), and which has been developed to fit typical aggregate data from the West.⁵ In terms of interpretation, \(\beta \) is a parameter that governs mostly the elevation of the weighting function, with higher values indicating a lower function. Since this indicates the weight assigned to the best outcome for gains, and the weight assigned to the worst outcome for losses, a higher value of \(\beta \) indicates increased probabilistic pessimism for gains, and increased probabilistic optimism for losses. Since we assume utility to be linear, we can directly interpret this parameter to indicate risk aversion for gains, and risk seeking for losses on average over the probability spectrum. For \(\alpha =1\), the parameter indeed represents standard risk aversion, with \(\beta \ge 1\) indicating risk aversion and \(\beta \le 1\) risk seeking. The parameter \(\alpha \) governs the slope of the probability weighting function, with \(\alpha =1\) indicating linearity of the weighting function (the EUT case), and \(\alpha <1\) representing the typical case of probabilistic insentivity.

The model considered so far is fully deterministic, assuming that subjects know their preferences perfectly well and execute them without making mistakes. It also assumes that we can capture such preferences perfectly in our model. Both assumptions seem untenable, especially in a development setting such as ours. We, thus, abandon this restrictive assumption and introduce an explicit stochastic structure. Given our setup, the certainty equivalent of a given prospect i we observe, \(ce_i\) will, thus, be equal to the certainty equivalent for the same prospect calculated from our model, \(\hat{ce}_i\), plus some error term, or \(ce_i=\hat{ce}_i+\epsilon _i\). We assume this error to be normally distributed, \(\epsilon _i\sim (0,\sigma _i^2)\), which allows for the errors to be serially correlated (see Train 2009). We can now express the probability density function \(\psi (.)\) for a given subject n as followswhere \(\phi \) is the standard normal density function, and \(\theta =\{\lambda ,\alpha ^j,\beta ^j,\}\) indicates the vector of parameters to be estimated. The subscript n to the parameter vector \(\theta \) indicates that we will let the parameters depend linearly on the observable characteristics of decision makers, such that \(\hat{\theta }=\hat{\theta }_k+\beta X\), where \(\hat{\theta }_k\) is a vector of constants and X represents a matrix of observable characteristics of the decision maker.⁶ Finally, \(\sigma \) indicates a so-called Fechner error (Hey and Orme 1994). The subscripts emphasize that we are allowing for three different types of heteroscedasticity, whereby n indicates as usual the observable characteristics of the decision maker, j indicates the decision domain (gains vs. losses), and i indicates that we allow the error term to depend on the specific prospect, or rather, on the difference between the high and low outcome in the prospect, such that \(\sigma _i=\sigma |x_i-y_i|\) (see Bruhin et al. 2010). For mixed prospects, we adopt the error term for losses, since only losses vary in the mixed choice list.

These parameters can now be estimated by maximum likelihood procedures. To obtain the overall likelihood function, we need to take the product of the density functions above across prospects and decision makers:where \(\theta \) is the vector of parameters to be estimated such as to maximize the likelihood function. Taking logs, we obtain the following log-likelihood function:We estimate this log-likelihood function in Stata using the Broyden–Fletcher–Goldfarb–Shanno optimization algorithm. Errors are always clustered at the subject level.

Table 2 shows a regression comparing the parameter estimates for farmers to those of American and Vietnamese students. The student data were obtained using the same experimental tasks as the ones used with the farmers and stakes are the same in terms of PPP. The American student data are borrowed from L’Haridon and Vieider (2018), who amongst other things report parameter estimates for the same model for students across 30 countries, and are meant to relate our current data to the results of that paper.⁷ In this sense, American students are meant to proxy for Western populations more in general—a point to which we will return shortly.

The regression controls for the sex of the respondent to address concerns that differences may be driven by gender effects that are often found for risk (Croson and Gneezy 2009), but the differences found are stable to dropping the demographic controls. Adding additional controls is difficult, as the control variables obtained for the student and farmer subject pools are generally not comparable. An exception to this is age. However, given that all our students are very young compared to the farmer sample, adding age results in high degrees of collinearity with the student dummies. We find that women are less sensitive to probabilistic change for both gains and losses, as well as less risk tolerant for losses. This is in line with the results reported by L’Haridon and Vieider (2018), who find a gender effect mostly on probabilistic sensitivity using a sample of almost 3000 students from 30 countries.

We find both student groups to be significantly more sensitive to probabilistic change than farmers, as indicated by the larger \(\alpha \) parameter. This holds for both gains and losses, and is consistent with the interpretation of the sensitivity parameter as a proxy for rationality or numeracy (Tversky and Wakker 1995; Wakker 2010). Students also show less noise in their decisions compared to the farmers. The farmers are slightly (but not significantly) less risk tolerant than Vietnamese students for gains, as indicated by the negative coefficient for the \(\beta ^+\) parameter. The farmers are, however, significantly more risk tolerant than the American students for both gains and losses (see Fig. 2). Indeed, American students exhibit decision patterns as they have typically been estimated in the West (see again L’Haridon and Vieider 2018), with a \(\beta ^+\) parameter slightly larger than one.⁸ For loss aversion, we find again that our farmers are intermediate between the two student populations, although none of the differences are significant.

Figure 4 shows the three risk-preference functions for gains together with the non-parametric data points (for losses see supplementary materials).⁹ The estimated functions can be seen to trace the nonparametric data closely. The risk-preference function for farmers is more elevated than the one of the American students up to and including \(p=\nicefrac {5}{8}\), and becomes very similar and somewhat lower for the two highest probability levels, respectively. Compared to Vietnamese students, the risk-preference functions are very similar up to at least \(p=\nicefrac {3}{8}\), after which the two functions start to diverge, as reflected in the farmers’ lower probabilistic sensitivity.

The comparison results may appear surprising, given a large number of studies showing risk aversion as the prevalent pattern in decisions under risk (we will discuss this point further below). To investigate the stability of these findings—and to be able to discuss average risk preferences over all choices with one simple measure—we now take the risk premium, given by EV\(\,-\,\)CE, averaged over all gain prospects (in PPP Euros; PPP US Dollars obtain by multiplication with 1.2).¹⁰ We here concentrate on gains—an equivalent analysis for losses is shown in the supplementary materials. Farmers have a significantly lower risk premium than American students (\(z=-2.07\), \(p=0.039\), two-sided Mann–Whitney test), and a (non-significantly) larger risk premium than Vietnamese students (\(z=1.54, p=0.123\)). While American students are significantly risk averse (\(z=4.67, p<0.001\); two-sided Wilcoxon signed-rank test), the farmers are on average risk neutral (\(z=0.23\), \(p=0.815\)), and Vietnamese students are on average risk seeking (\(z=-2.38, p=0.017\)). These findings are in no way unique to American students, whom we use as a typical exponent of Western subject pools. L’Haridon and Vieider (2018) indeed show a strong positive correlation between risk aversion and GDP per capita using students subject pools from 30 different countries over a wide range of income levels, and Vieider et al. (2018) report results for a general population sample from Ethiopia that is consistent with the evidence obtained for students.

So far, we have only considered aggregate preferences. The next step will be to look into individual characteristics and their correlation with risk preferences. In particular, we are interested in income, as well as measures of wealth. Table 3 shows the results of a regression of risk preferences on income per capita, education, and the age of the respondent (z-values are used for age, education, and income; using discrete categories for education does not change our results). Our subject pool is reduced to 197 subjects, since for the remaining subjects one of the observable characteristics is not reported.

We start by looking at the elevation of the risk preference function. For both gains and losses, we find risk tolerance to increase in income, as indicated by the negative coefficient for \(\beta ^+\) and the positive coefficient for \(\beta ^-\) (since for losses we transform probabilities attached to the worst outcome, following the going convention; see Wakker 2010). Loss aversion is also found to decrease in income. Probabilistic sensitivity for both gains and losses decreases with age. Given that probabilistic sensitivity is often taken to be an indicator of rationality or cognitive ability, this corresponds well to what we would expect. Also, the result corresponds closely to the results reported by L’Haridon and Vieider (2018), who find sensitivity to decrease in age and increase in grade point average. It is also in general agreement with findings by Choi et al. (2014), who found violations of rationality principles to increase with age and decrease with education and income (the latter effect is not significant in our data). Loss aversion is also found to increase in education. This is contrary to the findings of Gächter et al. (2010), but in agreement with the findings by von Gaudecker et al. (2011).

An important issue is whether our findings are indeed driven by income, and not by wealth. To capture wealth levels, we use the first two components from a principal component analysis into which all variables capturing wealth in our data set are entered, such as size and type of house, access to running water, sanitation facilities, motorcycles owned, ownership of TV or fridge, etc. (Filmer and Pritchett 2001). Table 4 reproduces the results from Table 3 controlling for these wealth indicators.¹¹ The wealth controls do not show any significant effects. This goes against the traditional assumption of risk aversion decreasing in wealth. The effect of income, however, only results reinforced from the introduction of wealth controls.

We, thus, find clear evidence of a negative correlation between risk aversion and income in our data. Such a negative correlation has frequently been reported in Western population samples (e.g., Dohmen et al. 2011; Donkers et al. 2001; Hopland et al. 2016), although sometimes it has been found only for a subset of the risk-preference measures used (for instance, von Gaudecker et al. 2011, and Booij et al. (2010) both only found the effect for loss aversion in representative Dutch samples, but not for utility curvature over gains), and at least one study found an effect going in the opposite direction (Harrison et al. 2007). A similar negative relationship with income or income proxies and risk aversion has also been found repeatedly in developing countries (Liebenehm and Waibel 2014; Vieider et al. 2018; Yesuf and Bluffstone 2009). Other studies, especially in developing countries, have not found any correlation between risk preferences and measures of well-being (Binswanger 1980; Cardenas and Carpenter 2013).