Risk aversion in the small and in the large: Calibration results for betweenness functionals

Abstract

A reasonable level of risk aversion with respect to small gambles leads to a high, and absurd, level of risk aversion with respect to large gambles. This was demonstrated by Rabin (Econometrica 68:1281–1292, 2000) for expected utility theory. Later, Safra and Segal (Econometrica, 2008) extended this result by showing that similar arguments apply to many non-expected utility theories, provided they are Gâteaux differentiable. In this paper we drop the differentiability assumption and by restricting attention to betweenness theories we show that much weaker conditions are sufficient for the derivation of similar calibration results.

Appendices

Appendix 1

Proof of Theorems 1 and 2

Let $\mathcal{I}$ be the indifference set of V through δ _w and let u(·;δ _w ) be one of the increasing vNM utilities obtained from $\mathcal I$ (note that all these utilities are related by positive affine transformations). Obviously, $\mathcal{I}$ is also an indifference set of the expected utility functional defined by $U(F)=\int u(x;\delta_w)dF(x)$. By monotonicity, V and U increase in the same direction relative to the indifference set $\mathcal{I}$.

We show first that u(·;δ _w ) is concave. Suppose not. Then there exist z and H such that z = E[H] and U(H) > u(z;δ _w ). By monotonicity and continuity, there exist p and z′ such that (z − w)(z′ − w) < 0 and $(z,p;z',1-p)\in \mathcal{I}$. As U(H) > u(z;δ _w ), it follows that (H,p;z′,1 − p) is above $\mathcal{I}$ with respect to the expected utility functional U. Therefore, (H,p;z′,1 − p) is also above $\mathcal{I}$ with respect to V, which is a violation of risk aversion, since (H,p;z′,1 − p) is a mean preserving spread of (z,p;z′,1 − p).

By betweenness, $V(x,1) > V(x-\ell,\frac12; x+g,\frac12)$ implies that, for all ε > 0, $V(x,1) > V(x,1-\varepsilon; x-\ell,\frac\varepsilon 2; x+g,\frac\varepsilon 2)$. Using the local utility at δ _x we obtain

$$ \label{e:31.1} {\textstyle u(x;\delta_x) > \displaystyle\frac12u(x-\ell;\delta_x) + \frac12 u(x+g;\delta_x)} $$

(3)

Theorem 1

Assume $\neg$H1 or $\neg$H2. We show that for every x ∈ [a,b], $u(x;\delta_a) > \frac12u(x- \ell;\delta_a) + \frac12 u(x+ g;\delta_a)$. Equation 3 holds for x = a, and $\neg$H1 implies

$$ \label{e:33.1} {\textstyle u(x;\delta_a) > \displaystyle\frac12u(x-\ell;\delta_a) + \frac12 u(x+g;\delta_a)} $$

(4)

Also, since $x\geqslant a$, $\neg$H2 implies that the local utility u(·;δ _a ) is more concave than the local utility u(·;δ _x ) and hence (4).

By Fact 1, it now follows that the expected utility decision maker with the vNM function u(·;δ _a ) satisfies u(a;δ _a ) > pu(a − L;δ _a ) + (1 − p) u(a + G;δ _a ) and the lottery (a − L,p;a + G,1 − p) lies below the indifference set $\mathcal{I}$. Therefore,

$$ {\textstyle V(a,1)> V(a-L,p;a+ G,1-p)} $$

Theorem 2

Assume H1 or H2. Here we show that for every x ∈ [a,b], $u(x;\delta_b) > \frac12u(x-\ell;\delta_b) + \frac12 u(x+g;\delta_b)$. Equation 3 holds in particular for x = b, and therefore H1 implies that for $x \leqslant b$,

$$ \label{e:31.2} {\textstyle u(x;\delta_b) > \displaystyle\frac12u(x-\ell;\delta_b) + \frac12 u(x+g;\delta_b)} $$

(5)

Also, since $x \leqslant b$, H2 implies that the local utility u(·;δ _b ) is more concave than the local utility u(·;δ _x ). Hence starting at (3) and moving from δ _x to δ _b , H2 implies (5).

By Fact 2 it now follows that the expected utility decision maker with the vNM function u(·;δ _b ) satisfies u(b;δ _b ) > pu(b − L;δ _b ) + (1 − p) u(b + G;δ _b ) and the lottery (b − L,p;b + G,1 − p) lies below the indifference set $\mathcal{I}$. Therefore,

$$ {\textstyle V(b,1)> V(b-L,p;b+G,1-p)}\quad \quad \square $$

Proof of Theorem 3

For every x and z such that $w+c \geqslant x > w > z \geqslant w-\bar L$ there is a probability q such that X = (x,q;z,1 − q) satisfies V(X) = V(w,1). By (ℓ,g) bi-stochastic B3,

$$ \label{e:7.2} {\textstyle V(x,q;z,1-q) > V\left(x-\ell,\displaystyle\frac q2; x+g, \frac q2; z-\ell,\frac{1-q}2; z+g,\frac{1-q}2\right)} $$

(6)

Denote the distribution functions of the two lotteries in (6) by F and F′, respectively. Betweenness implies that the local utilities at δ _w and at F are the same (up to a positive linear transformation). Hence, by using the local utility u(·;δ _w ), we obtain,

$$ \begin{array}{lll} \label{e:7.3} & & { q\left[u(x;\delta_w) - \displaystyle\frac12(u(x-\ell;\delta_w) + u(x+g;\delta_w))\right] }\\ & &{\kern12pt} + { (1-q)\left[u(z;\delta_w) - \displaystyle\frac12(u(z-\ell;\delta_w) + u(z+g;\delta_w))\right] > 0} \end{array} $$

(7)

Suppose there are $w\!+\!c\! >\! x^*\! >\! w\! >\! z^* \!> \!w\!-\!\bar L$ such that $u(x^*;\delta_w) \!\leqslant\! \frac12[u(x^*\!-\!\ell;\delta_w) \!+\! u(x^*\!+\!g;\delta_w)]$ and $u(z^*;\delta_w) \leqslant \frac12[u(z^*\!-\!\ell;\delta_w) \!+\! u(z^*\!+\!g;\delta_w)]$. Then inequality (7) is reversed, and by betweenness, the inequality at (6) is reversed; a contradiction to bi-stochastic B3. Therefore, at least one of the following holds.

• 1. For all x ∈ [w,w + c],

  $$ {\displaystyle u(x;\delta_w) > \frac12[u(x-\ell;\delta_w) + u(x+g;\delta_w)]} $$

  and, similarly to the Proof of Theorem 1 (replace a with w in (4)), betweenness implies that V(w,1) > V(w − L,p;w + G,1 − p) and the first claim of the theorem is satisfied.

• 2. For all $x \in [w-\bar L,w]$,

  $$ {\displaystyle u(x;\delta_w) > \frac12[u(x-\ell;\delta_w) + u(x+g;\delta_w)]} $$

  and, using an argument similar to that of the Proof of Theorem 2 (replace b with w in (5)), betweenness implies $V(w,1)> V(w-g-\bar L,\bar p;w-g+\bar G,1-\bar p)$ and the second claim of the theorem is satisfied. □

Appendix 2

We first show that Gul’s (1991) disappointment aversion functional is not Gâteaux differentiable.⁵ Consider lotteries of the form (x,p;y,1 − p − q;z,q) where x < y < z are constant and a disappointment aversion functional V(p,q) with u(x) = 0 and u(z) = 1. From (1) and (2) in Gul (1991) it follows that

$$ V(p,q) = \left \{ \begin{array}{ll} \displaystyle\frac{(1-p-q)u(y)+q}{1+\beta p} & q \leqslant \displaystyle\frac{(1+\beta) p u(y)}{1-u(y)}\\\\ \displaystyle\frac{(1+\beta)(1-p-q)u(y)+q}{1+ \beta - \beta q} & q > \displaystyle\frac{(1+\beta) p u(y)}{1-u(y)} \end{array} \right. $$

Along the line p + q = 1 we obtain

$$ V(p,q)=\frac{1-p}{1+\beta p} $$

hence along this line, V is differentiable. In particular, the derivatives at $\left(\frac{1-u(y)}{1+\beta u(y)}, 1- \frac{1-u(y)}{1+\beta u(y)} \right)$ in the directions (1,0) and (0,1) have the same absolute value and opposite signs. However, along the line $p+q=\frac12$, we obtain

$$ V(p,q)=\left\{ \begin{array}{ll} \displaystyle\frac{u(y)+1-2p}{2+2\beta p} & p \geqslant \displaystyle\frac{1-u(y)}{2+2\beta u(y)}\\\\ \displaystyle\frac{(1+\beta)u(y)+1-2p}{2+2\beta p+\beta} & p < \displaystyle\frac{1-u(y)}{2+2\beta u(y)} \end{array} \right. $$

It is easy to verify that the derivatives of V at $\left(\frac{1-u(y)}{2+2\beta u(y)}, \frac12- \frac{1-u(y)}{2+2\beta u(y)} \right)$ in the directions $(\frac12,0)$ and $(0,\frac12)$ do not have the same absolute value. As V is differentiable along the line p + q = 1 but not along the line $p+q=\frac12$, it follows that neither V, nor any increasing transformation of V is Gâteaux differentiable.

Next we show that hypotheses H1, H2, and their opposites do not imply Gâteaux differentiability. It is easy to construct a weighted utility functional W (Chew 1983) satisfying H1 and H2. Define a functional V which is equal to W above indifference curve $\cal I$, and below $\cal I$ it is expected utility with $\cal I$ being one of its indifference curves. Similarly to the above analysis, this functional is not Gâteaux differentiable, but as expected utility and W satisfy H1 and H2, so does V.