Risk preferences and changes in background risk

Abstract

We present two theorems that yield necessary and sufficient conditions for first- and second-degree stochastic dominance deteriorations of background risk to increase risk aversion with respect to foreground risk. We require that any change in a foreground risk that is undesirable remains so after a background risk changes in a way that is either unfair, undesirable in the sense of reducing expected utility, or undesirable in the sense of increasing expected marginal utility. Our results thus characterize utility functions that are, respectively, vulnerable, proper, or standard with respect to changes in background risk.

Appendix: Proofs of theorems

In this Appendix we provide proofs of the results stated in the text.

Proof of Lemma 1

We wish to show that R ₂(x) ≥ R ₁(x) for all x is sufficient in order that $Eu(\tilde{x}_2 +\tilde{y}_2) \leq Eu(\tilde{x}_1+\tilde{y}_2)$ for all $\tilde{x}_2$ that induce an SSD spread in υ ₁(x), and that this condition is necessary given that $\tilde{x}_1$ has full support. We have

$$\begin{array}{*{20}c} {Eu{\left( {\widetilde{x}_{2} + \widetilde{y}_{2} } \right)} - Eu{\left( {\widetilde{x}_{1} + \widetilde{y}_{2} } \right)} = {\int {v_{2} {\left( x \right)}d{\left[ {F_{2} {\left( x \right)} - F_{1} {\left( x \right)}} \right]}} }} \\ { = - {\int {\frac{{v^{\prime }_{2} {\left( x \right)}}}{{v^{\prime }_{1} {\left( x \right)}}}v^{\prime }_{1} {\left( x \right)}{\left[ {F_{2} {\left( x \right)} - F_{1} {\left( x \right)}} \right]}dx} }} \\ { = {\int {\frac{\partial }{{\partial x}}{\left( {\frac{{v^{\prime }_{2} {\left( x \right)}}}{{v^{\prime }_{1} {\left( x \right)}}}} \right)}{\int_{}^x {v^{\prime }_{1} {\left( s \right)}{\left[ {F_{2} {\left( s \right)} - F_{1} {\left( s \right)}} \right]}dsdx} }} }} \\ \end{array} $$

(9)

after applying integration by parts twice. The second integral on the last line is nonnegative for all x since the change from F ₁(x) to F ₂(x) induces an SSD deterioration in the distribution of utility $v_1(\tilde{x}).$ Hence, the entire expression is nonpositive if

$$ \frac{\partial}{\partial x}\Bigl( \frac{\upsilon'_2(x)}{\upsilon'_1(x)} \Bigr)=\frac{\upsilon'_2(x)}{\upsilon'_1(x)}[R_1(x)-R_2(x)] $$

(10)

is nonpositive for all x. Thus, R ₂(x) ≥ R ₁(x) for all x is sufficient for 9 to be nonpositive. This condition is also necessary, since were it to fail at an interior point $\hat{x}$, it would have to fail in an entire neighborhood of $\hat{x}$. Given that F ₁(x) being strictly monotonic implies that it must have mass in this neighborhood, we could perform a local mean utility preserving spread there, and so choose F ₂(x) to coincide with F ₁(x) outside the neighborhood, with the F ₂(x)-weighted marginal utility integral strictly exceeding the one weighted by F ₁(x) only within the neighborhood. With R ₂(x) < R ₁(x) in this neighborhood, 9 would be made positive, contrary to our hypothesis. □

Rather than examine vulnerability, properness, and standardness separately, we prove Theorems 1 and 2 by covering the three cases simultaneously. We accomplish this by introducing the transformation function ϕ = ϕ _m (y) , and three alternative representations, ϕ _v (y) = y, $\varphi_p(y) = \int u(x+y)dF_1(x)$, and $\varphi_s(y) \!=\! -\!\int u'(x\!+\!y)dF_1(x),$ to characterize decision makers who are, respectively, vulnerable, proper, and standard. In each instance, ϕ _m (y) is concave and strictly increasing, so a strictly increasing inverse function y(ϕ) exists. Replacing x + y with x + y(ϕ), we work with background risks G _j (ϕ) and derived utility functions $\upsilon_j(x)=\int u(x + y(\varphi))dG_j(\varphi)$. Thus, for example, when ϕ = ϕ _p (y), any change in background risk y that induces a second-degree spread of G ₁(ϕ) results in a second-degree spread in the distribution for $\int u(x+y)dF_1(x)$ and any such change increases the degree of risk aversion for υ(x) if and only if u exhibits properness. Note that in Theorem 1, the first-degree case, the particular transformation turns out not to make a difference, so one should be able to dispense with such transformations by a prior argument. However, this is certainly not true in the second-degree case, and we prefer to use a general approach for both cases to highlight the sources of any differences in the restrictions that characterize vulnerability, properness, and standardness.

Proof of Theorem 1

The proof for vulnerability, when ϕ = ϕ _v (y), is provided by Eeckhoudt, Gollier and Schlesinger (1996) in the proof of their Proposition 2. The sufficiency proof applies as well to properness, when ϕ = ϕ _p (y), and to standardness, when ϕ = ϕ _s (y), since u and − u′ are both strictly increasing, so that the set of FSD deteriorations in the distribution of the background risk is the same in each of the three cases. For example, in the case of properness, an FSD deterioration requires $Pr[u(x+\tilde{y}_1)<U]\le Pr[u(x+\tilde{y}_2)<U]\ \forall U$, which is equivalent to $Pr[\tilde{y}_1<t]\le Pr[\tilde{y}_{2}<t]\ \forall t$ with t ≡ u ^− 1(U) − x, which is the requirement for an FSD deterioration in the distribution of the background risk.

Similarly, the necessity part of the Eeckhoudt, Gollier and Schlesinger (1996) proof can be adapted to show that if u is, say, proper and 2 were not to hold, then at some $(\hat{x}, \hat{y}$, $\hat{y}')$ there would be a G ₁(ϕ) and a G ₂(ϕ) such that 1 would be violated at $\hat{x}$. But Lemma 1 implies that condition 1 must always hold, given that u is assumed to be proper with respect to all initial distributions F ₁(x), and hence with respect to strictly monotonic ones. □

Proof of Theorem 2

We first state and prove a lemma that relates mean-preserving spreads to inequality 1 indicating greater risk aversion. Just as we defined the derived utility function υ _j (x) induced by background risk G _j (y), we can define the utility function induced by foreground risk, $\mu_i(y)\!=\! Eu(\tilde{x}_i\!+\!y) \!=$ $\int u(x+y)dF_i(x),$ and its associated measures, $\hat{R}_i(y)= -\mu_i''(y)/\mu_i'(y)$ for absolute risk aversion, and $\hat{P}_i(y) = -\mu_i'''(y)/\mu_i''(y)$ for absolute prudence.□

Lemma 2

(mean-preserving spreads)

(vulnerable case) Condition 1 is satisfied for all mean-preserving spreads of any background risk G ₁(ϕ _v ) if and only if

$$ t(x+y)\ge r(x+y') \ \ \ f\!or\ all\ x,y,y'. $$

(11)

(proper case) Given any initial foreground risk F ₁(x), condition 1 is satisfied for all mean-preserving spreads of any background risk G ₁(ϕ _p ) if and only if

$$p{\left( {x + y} \right)}{\left[ {t{\left( {x + y} \right)} - \hat{R}_{1} {\left( y \right)}} \right]} \geqslant r{\left( {x + y^{\prime } } \right)}{\left[ {p{\left( {x + y} \right)} - \hat{R}_{1} {\left( y \right)}} \right]}\;\;\;for\;all\;x,y,y^{\prime } .$$

(12)

(standard case) Given any initial foreground risk F ₁(x), condition 1 is satisfied for all mean-preserving spreads of any background risk G ₁(ϕ _v ) if and only if

$$ p(x+y)[t(x+y) - \hat{P}_1(y)] \ge r(x+y')[p(x+y) - \hat{P}_1(y)]\ \ \ f\!or\ all\ x,y,y'. $$

(13)

Proof of Lemma 2

Assume that G ₂(ϕ) is an MPS of G ₁(ϕ). This change results in greater risk aversion if and only if inequality 1 holds, or equivalently,

$$-\frac{\upsilon''_2(x)}{\upsilon_2'(x)} \ge -\frac{\upsilon''_1(x)}{\upsilon_1'(x)}\ f\!or\ all\ x.$$

(14)

Multiplying both sides of 14 by υ′₂(x) and then adding υ′′₁(x) yields the equivalent inequality

$$[\upsilon''_1(x) - \upsilon''_2(x)]\ge - \frac{\upsilon''_1(x)}{\upsilon_1'(x)}[\upsilon'_2(x) - \upsilon'_1(x)]. $$

(15)

From the definition of υ _i (x), we find that 15 is equivalent to

$$ -\!\int\! u''(x \!+\!y)d[G_2(\varphi) \!-\!G_1(\varphi)] \!\ge\!\! -\frac{\upsilon''_1(x)}{\upsilon_1'(x)}\int u'(x \!+\!y)d[G_2(\varphi)\!-\!G_1(\varphi)], $$

(16)

with y = y(ϕ).

We show that

$$ -\frac{u'''(x\!+\!y)}{u''(x\!+\!y)}\Bigl[-\frac{u''''(x\!+\!y)}{u'''(x\!+\!y)} \!+\! \frac{\varphi_m''(y)}{\varphi_m'(y)}\Bigr]\!\ge\! -\frac{u''(x\!+\!y')}{u'(x\!+\!y')} \Bigl[\!-\frac{u'''(x\!+\!y)}{u''(x\!+\!y)} \!+\! \frac{\varphi_m''(y)}{\varphi_m'(y)}\Bigr]\notag f\!or\ all\ x,y,y'$$

(17)

is necessary and sufficient for 16 and hence 1, where m is either v, p, or s.

Applying integration by parts twice to the integrals on both sides of inequality 16 requires differentiating u′(x + y(ϕ)) and u′′(x + y(ϕ)) twice with respect to ϕ. For the first derivative of u′, we obtain

$$ u''(x+y)y_\varphi = u''(x+y)/\varphi'_m(y),$$

(18)

where it is again understood that y = y(ϕ), and for the second derivative we obtain

$$u'''(x+y)y_\varphi^2+u''(x+y)y_{\varphi\varphi}=\Bigl[\frac{u'''(x+y)}{u''(x+y)}-\frac {\varphi''_m(y)}{\varphi'_m(y)}\Bigr]u''(x+y)y_\varphi^2. $$

(19)

For the derivatives of u′′, we obtain expressions similar to 18 and 19, but with u being differentiated one more time. Thus, when we carry out the integration by parts and combine the two sides, we arrive at

$$\int T(x,\varphi)u''(x+y)y_\varphi^2 \Bigl\{ \int^\varphi [G_2(\omega)-G_1(\omega)]d\omega \Bigr\} d\varphi \le 0, $$

(20)

where

$$\begin{array}{*{20}c} {T(x,\varphi) = -\frac{u'''(x+y)}{u''(x+y)}\Bigl[-\frac{u''''(x+y)}{u'''(x+y)} + \frac{\varphi_m''(y)}{\varphi_m'(y)}\Bigr] +\frac{\upsilon''_1(x)}{\upsilon_1'(x)} \Bigl[-\frac{u'''(x+y)}{u''(x+y)}+ \frac{\varphi_m''(y)}{\varphi_m'(y)}\Bigr]} \\ { \equiv T_{1} {\left( {x,\varphi } \right)} + \frac{{v^{{\prime \prime }}_{1} {\left( x \right)}}}{{v^{\prime }_{1} {\left( x \right)}}}T_{2} {\left( {x,\varphi } \right)}.} \\ \end{array} $$

(21)

Here, the second line serves to define T ₁ and T ₂. Since G ₂(ϕ) is an MPS of G ₁(ϕ), the integral within braces on the left-hand side of 20 is always nonnegative. Additionally, $u''(x+y)y_\varphi^2$ is negative. Hence, inequality 20 holds, and therefore inequality 16 holds, if and only if T is nonnegative for all x and ϕ.

It is straightforward to establish sufficiency of 17. Let $\overline{M}(x)$ and $\underline{M}(x)$ denote, respectively, the maximum and minimum values of − u′′(x + y)/u′(x + y) over y(ϕ). Observe that

$$-\frac{\upsilon''_1(x)}{\upsilon_1'(x)} = - \int \frac{u''(x+y)}{u'(x+y)}\frac{u'(x+y)G_1'(\varphi)}{\int u'(x+y)dG_1(\varphi)}d\varphi \le \overline{M}(x),$$

(22)

where the inequality reflects the fact that the second ratio in the integrand can be interpreted as a probability density. The same logic yields

$$\overline{M}(x)\ge - \frac{\upsilon''_1(x)}{\upsilon_1'(x)} \ge \underline{M}(x).$$

(23)

For any values of x and ϕ such that T ₂(x,ϕ) ≥ 0, we have

$$ T_1(x,\varphi)\ge\overline{M}(x)T_2(x,\varphi) \ge -\frac{\upsilon''_1(x)}{\upsilon_1'(x)}T_2(x,\varphi)$$

(24)

by combining condition 17 with the first inequality of 23. These inequalities then imply that T is nonnegative. Alternatively, for any values of x and ϕ such that T ₂(x,ϕ) < 0, we have

$$ T_1(x,\varphi)\ge\underline{M}(x)T_2(x,\varphi)\ge -\frac{\upsilon''_1(x)}{\upsilon_1'(x)}T_2(x,\varphi)$$

(25)

by combining condition 17 and the second inequality of 23. These inequalities now imply that T is again nonnegative. Thus, condition 17 is sufficient for any MPS G ₂(ϕ) of background risk G ₁(ϕ) to increase risk aversion with respect to foreground risk, that is, for R ₂(x) ≥ R ₁(x) for all x.

The proof of necessity of 17 then follows a line of argument parallel to that used in the proof of necessity for Theorem 1. Finally, note that 17 is equivalent to 11, 12, and 13 when m is equal to v, p, and s, respectively. □

Completion of proof of theorem 2

For vulnerability, it suffices to observe that condition 4 is equivalent to 11. The sufficiency of condition 6 for standardness follows from the observation made by Gollier (2001), Eq. 8.4, p. 115, that $\hat{P}(y) = E[ p(\tilde{x}+y)u''(\tilde{x}+y)/E[u''(\tilde{x}+y)]]$. Thus, we have $p(\underline{x}\!+\!y) \!=\! p(\underline{x}\!+\!y)$ $E[u''(\tilde{x}\!+\!y)/\!E[u''(\tilde{x}\!+\!y)]]\!\le\! \hat{P}_1(y)\!\le\! p(\overline{x}\!+\!y)E[u''(\tilde{x}\!+\!y)/\!E[u''(\tilde{x}\!+\!y)]] \!=\!\! p(\overline{x}\!+\!y)$, where $\underline{x}$ minimizes p(x + y) over the range of x and $\overline{x}$ maximizes it. Thus we obtain from 6 that $p(x\!+\!y)[t(x\!+\!y)\!-\!\hat{P}_1(y)]\!\ge\! p(x\!+\!y)[t(x\!+\!y)\!-\!p(\overline{x}\!+\!y)] \!\ge\! r(x\!+\!y')[p(x\!+\!y)\!-\! p(\underline{x}\!+\!y)] \!\ge\! r(x\!+\!y')[p(x\!+\!y)\!-\! \hat{P}_1(y)]$, which implies 13. By Lemma 2, 13 implies condition 1 and then Lemma 1 allows us to conclude that u satisfies standardness with respect to mean-preserving spreads. The sufficiency of condition 5 for properness follows in a similar manner from $\hat{R}(y) = E[ r(\tilde{x}+y)u'(\tilde{x}+y)/E[u'(\tilde{x}+y)]]$.

To establish the necessity of condition 6 for standardness, assume that u is standard with respect to mean-preserving spreads and suppose that 6 is violated at $(\hat{x},\hat{x}',\hat{y},\hat{y}')$. Lemma 2 states that condition 13 is necessary and sufficient for condition 1 always to hold, given any initial foreground risk F ₁(x). It then suffices to choose F ₁(x) strictly monotonic, but with almost all of its mass in a neighborhood of $\hat{x}'$, so that $P_1(y)\approx p(\hat{x}' + y)$. Thus, condition 13 would be violated at $(\hat{x},\hat{y},\hat{y}')$ for the chosen F ₁(x), implying, by Lemma 2, that condition 1 would not always hold. But this would be in direct contradiction with Lemma 1, which implies that condition 1 must always hold, given that F ₁(x) is strictly monotonic and u is standard. Hence, the supposition that 13 can be violated when u is standard must be false. A similar argument establishes that 12 is necessary for properness. □

Proof of Corollary 1

Any SSD deterioration can be decomposed into an MPS plus an FSD deterioration, as shown by Machina and Pratt (1997). Hence sufficiency of 2 and 6 for standardness is then an immediate consequence of Theorems 1 and 2. Necessity of 2 follows since an FSD deterioration is a particular kind of SSD deterioration, while necessity of 6 similarly follows because an MPS is also a particular kind of SSD deterioration. The same arguments apply to properness and vulnerability. □

Proof of Corollary 2

Background Ross DARA 2 is one of the necessary and sufficient conditions for standardness stated in Corollary 1. Since Ross DAP implies DAP, we have t(x + y) ≥ p(x + y) for all x, y, and therefore

$$ t(x+y)- p(x'+y) \ge p(x+y)- p(x'+y) \ f\!or\ all \ x, x',y.$$

(26)

Moreover, Ross DAP implies that the left-hand side is nonnegative. Since, by Ross DARA 2, p(x + y) ≥ r(x + y′) ≥ 0 for all x, y, y′, we can multiply the left-hand side of 26 by p(x + y) and the right-hand side by r(x + y′), preserving the inequality and arriving at condition 6 for standardness.

The necessity of DAP for standardness follows from setting x = x′ in 6. □

Proof of Corollary 3

From DAP we have t(x + y) ≥ p(x + y) for all x,y and therefore

$$ t(x+y)- r(x'+y) \ge p(x+y)- r(x'+y)\ f\!or \ all \ x,x',y.$$

(27)

Now, DAP and foreground Ross DARA 3 yield t(x + y) ≥ p(x + y) ≥ r(x′ + y), which implies that the left-hand side of 27 is nonnegative. The rest of the argument follows as in the Proof of Corollary 2 to arrive at condition 5 for properness. □

Proof of Corollary 4

DAP and Ross DARA 2 yield t(x + y) ≥ p(x + y) ≥ r(x + y′) for all x,y, y′, which implies condition 4 for vulnerability. □