Probabilidades de los escenarios

In this section, I analyze in more detail how non-parametric risk aversion is measured in the risk task and how this measure of risk aversion is related to a parametric estimate of constant relative risk aversion (CRRA). Consider Figure B.1, which presents a typical round in a risk task. The budget line is defined by two points:M_xandM_ythat represent the maximum number of tokens that can be allocated to accountsxandy, respectively. In this case, accountyis a cheaper account, since

My >Mx. PointC on the graph corresponds to a point of an equal allocation between accounts.

This allocation yields an amount ¯x=y¯regardless of an outcome. Only subjects with extremely high degrees of risk aversion would choose this point. PointArepresents a risk-neutral allocation, since this allocation yields the highest expected value. Moving from pointCto pointArepresents an increase in the expected value of a lottery while increasing its variance. A moderately risk- averse subject, therefore, would choose an allocation along the line segment AC, which in this example is point B at whichx0 tokens are allocated to account x, andy0 tokens are allocated to

accounty. Allocations on the line segmentCDare first-order stochastically dominated (FOSD) by all other allocations: moving fromC toDresults in a decrease in the expected value of a lottery while increasing its variance.

A B C D y0 ¯ y My 0 x0 x¯ Mx x y

Figure B.1: Geometry of the Risk Task

Intuitively, the closer the chosen allocation point B to the equal-allocation pointC, the more risk averse a subject is. On the flip side, the closer pointBto a risk-neutral allocation pointA, the less risk averse a subject is. Therefore, a subject’s risk aversion can be defined as the ratio of the lengths of line segmentsAB toAC. The resulting ratio should lie in the unit interval for subjects who observe FOSD, with a value of 0 meaning risk neutrality and a value of 1 meaning extreme risk aversion. A value of the ratio greater than 1 implies a violation of FOSD. Practically, the ratio can be found asx0/x¯,4where the equal allocation point is given by ¯x=y¯=MxMy/(Mx+My). The

4_{Provided thatxis a more expensive account. Ifxis a cheaper account, the ratio will be defined asy} 0/y¯.

non-parametric measure of risk aversion for a given subject is then defined as the average of the ratios in each round.

Figure B.2 (left panel) shows the distribution of this measure of risk aversion in the sample. The distribution has a spike around 1, which implies that many subjects have chosen allocations near the equal-allocation point. In fact, 21% of the subjects have estimated risk aversion within a range of 1±0.01. Some subjects can be classified as risk-neutral, which is evident by a small spike around 0. The majority of the subjects, 81%, satisfy FOSD, on average. If one allows for mistakes and classifies subjects as satisfying FOSD with risk aversion below 1.05 instead, then 97% of the subjects will satisfy this criterion.

0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Risk Aversion Density 0.00 0.05 0.10 0.15 0.20 0.25 -5 0 5 10 15 Logarithm of CRRA Density Kendall’sτ= 0.755 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Risk Aversion CRRA

Figure B.2: Risk Aversion in the Sample

To relate the non-parametric measure of risk aversion to a commonly used parametric estimate of CRRA, one would need to estimate the CRRA parameter from the data. Assume that subjects maximize expected utility and have a utility-of-money function of the formu(x) =x1−r/(1−r), where r is the CRRA parameter. One can show that an optimal allocation to account x,5 as a function of CRRA and design parameters M_x andM_y is given byx∗(r,M_x,M_y) =aM_xM_y/(M_x+ aMy), where a≡(Mx/My)1/r. Higher values of r imply that ais closer to 1, and therefore x∗ is

closer to ¯x. On the other hand, lower values of r lead to a chosen allocation being closer to the risk-neutral allocation. Knowing the closed-form solution for the optimal allocation then allows one to estimaterfor each subject using data on chosen allocations{x}iand on given budget lines

{M_x,M_y}i, in each roundi.

Figure B.2(center panel) shows the distribution of the logarithmof the estimated CRRA co- efficients in the sample. The logarithmic transformation is used so that the picture can fit in the subject with very high CRRA estimates. These are precisely the subjects who chose allocation near the equal-allocation point. The distribution has a spike at 0, which implies a special case of logarithmic utility for many subjects. Overall, the estimated CRRA coefficients are large.

FigureB.2(right panel) shows the relation between the non-parametric measure of risk aversion and estimated CRRA coefficients.6 There is a high positive association between the two measures, with Kendall’sτ = 0.755 (p−value<0.001). The relationship between the two measures is highly

non-linear and is characterized by a convex shape. Around the point where the non-parametric measure of risk aversion approaches 1 (marked by the vertical dashed line) the estimate of CRRA starts to approach infinity, as only subjects withr=∞would select the equal-allocation point.

5_{One only needs to determine an optimal allocation to one of the accounts, as it exactly pins down an optimal} allocation to the remaining account via the budget line equationy=−(M_y/Mx)x+My.