An interpretation of the condition for precautionary saving: the case of greater higher-order interest rate risk

Abstract

This paper shows that an increase in interest rate risk via (m, n)th-order stochastic dominance induces precautionary saving if, and only if, the measure of $(k+1)$th-degree relative risk aversion exceeds k for all $k=m,\ldots , n$. This result has the following interpretation. On the one hand, the measures of $(k+1)$th-degree relative risk aversion for all $k=m,\ldots , n$ capture the prudence effect with respect to a risk increase via (m, n)th-order stochastic dominance, which favors precautionary saving. On the other hand, the thresholds, $k=m,\ldots , n$, measure the elasticity of the change in the kth moment of future income with respect to saving. The adverse changes in higher moments of future income for all $k=m,\ldots , n$ when saving increases give rise to the risk aversion effect with respect to a risk increase via (m, n)th-order stochastic dominance, which limits precautionary saving. The necessary and sufficient condition for precautionary saving simply states that the prudence effect dominates the risk aversion effect, thereby making precautionary saving prevail.

Appendices

Appendix

A. Proof of Proposition 1

To show that statement (i) implies statement (ii), we differentiate Eq. (8) with respect to s and evaluate the resulting derivative at $s=0$ to yield

$$\begin{aligned} \phi '(0)= & {} -u'\big (w_{\circ }-s^{\ell }\big )+\frac{1}{1+\delta }\int _{a}^{b}u'(s^{\ell }x)x\mathrm{d}F(x) \nonumber \\= & {} \frac{1}{1+\delta }\int _{a}^{b}u'(s^{\ell }x)x\mathrm{d}[F(x)-G(x)] \nonumber \\= & {} \frac{(s^{\ell })^{-2}}{1+\delta }\bigg \{\sum _{k=m}^{n-1} (-s^{\ell })^{k+1}u^{(k)}(s^{\ell }b) \bigg [-s^{\ell }b\frac{u^{(k+1)}(s^{\ell }b)}{u^{(k)}(s^{\ell }b)}-k\bigg ]\big [F_{k+1}(b)-G_{k+1}(b)\big ]\nonumber \\&\quad +\int _a^b (-s^{\ell })^{n+1}u^{(n)}(s^{\ell }x)\bigg [-s^{\ell }x\frac{u^{(n+1)}(s^{\ell }x)}{u^{(n)}(s^{\ell }x)}-n\bigg ] [F_n(x)-G_n(x)]\mathrm{d}x\bigg \}, \nonumber \\ \end{aligned}$$

(A.1)

where the second equality follows from Eq. (6), and the last equality follows from applying integration by parts and $F_k(b)=G_k(b)$ for all $k=2,\ldots , m$. Since $F_k(b)\ge G_k(b)$ for all $k=m+1,\ldots , n$ and $F_n(x)\ge G_n(x)$ for all $x\in [a,b]$, it follows from Eq. (10) that the right-hand side of Eq. (A.1) is positive, rendering that $s^h>s^{\ell }$.

To show that statement (ii) implies statement (i), we suppose the contrary that there exist an integer, $k\in \{m,\ldots ,n\}$, and a point, $w'\in [s^{\ell }a,s^{\ell }b]$, such that Eq. (10) does not hold. By continuity, we have

$$\begin{aligned} -w\frac{u^{(k+1)}(w)}{u^{(k)}(w)}\le k, \end{aligned}$$

(A.2)

for all $w\in [w'-\varepsilon _1,w'+\varepsilon _2]$, where $\varepsilon _1$ and $\varepsilon _2$ are two small non-negative numbers such that $w'-\varepsilon _1\ge s^{\ell }a$ and $w'+\varepsilon _2\le s^{\ell }b$. Let $x_1=(w'-\varepsilon _1)/s^{\ell }$ and $x_2=(w'+\varepsilon _2)/s^{\ell }$. It follows that $a\le x_1<x_2\le b$. Construct F(x) such that $F(x)=G(x)$ for all $x\in [a,x_1]\bigcup [x_2,b]$, $F_j(x_2)=G_j(x_2)$ for all $j=2, \ldots ,k$, and $F_k(x)\ge G_k(x)$ for all $x\in [x_1,x_2]$. F(x) as such has more kth-degree risk than G(x) and thus F(x) is riskier than G(x) via (m, n)th-order stochastic dominance. In this case, Eq. (A.1) becomes

$$\begin{aligned} \phi '(0)=\frac{(s^{\ell })^{k-1}}{1+\delta } \int _{x_1}^{x_2} (-1)^{k+1}u^{(k)}(s^{\ell }x)\bigg [-s^{\ell }x\frac{u^{(k+1)}(s^{\ell }x)}{u^{(k)}(s^{\ell }x)}-k\bigg ] [F_k(x)-G_k(x)]\mathrm{d}x. \nonumber \\ \end{aligned}$$

(A.3)

Since $F_k(x)\ge G_k(x)$ for all $x\in [x_1,x_2]$, it follows from Eq. (A.2) that the right-hand side of Eq. (A.3) is non-positive, rendering that $s^h\le s^{\ell }$, a contradiction. Hence, Eq. (10) must hold for all $w\in [s^{\ell }a,s^{\ell }b]$ and for all $k=m,\ldots , n$.

B. Derivation of Eq. (18)

Applying kth-order Taylor expansions to $\mathrm{E}[u(s^{\ell }{\tilde{x}}^{\ell })]$ and $\mathrm{E}[u(s^{\ell }{\tilde{x}}^h)]$ around $x=x^{\circ }$ yields

$$\begin{aligned} \mathrm{E}\big [u(s^{\ell }{\tilde{x}}^{\ell })\big ]-\mathrm{E}\big [u(s^{\ell }{\tilde{x}}^h)\big ]\approx & {} \sum _{j=0}^k \frac{1}{j!}u^{(j)}(s^{\ell }x^{\circ }) \Big \{\mathrm{E}\big [(s^{\ell }{\tilde{x}}^{\ell }-s^{\ell }x^{\circ })^j\big ]-\mathrm{E}\big [(s^{\ell }{\tilde{x}}^h-s^{\ell }x^{\circ })^j\big ]\Big \}\nonumber \\= & {} \frac{1}{k!}u^{(k)}(s^{\ell }x^{\circ })\Delta _k(s^{\ell }), \end{aligned}$$

(A.4)

where the equality follows from $\mathrm{E}[({\tilde{x}}^{\ell })^j]=\mathrm{E}[({\tilde{x}}^h)^j]$ for all $j=1,\ldots , k-1$ and $\Delta _k(s^{\ell })=(s^{\ell })^k\{\mathrm{E}[({\tilde{x}}^{\ell })^k]-\mathrm{E}[({\tilde{x}}^h)^k]\}$. Likewise, we have

$$\begin{aligned} \mathrm{E}\big [u(sx^{\circ } +s^{\ell }{\tilde{x}}^{\ell })\big ]-\mathrm{E}\big [u(sx^{\circ }+s^{\ell }{\tilde{x}}^h)\big ] \approx \frac{1}{k!}u^{(k)}\big ((s^{\ell }+s)x^{\circ }\big )\Delta _k(s^{\ell }), \end{aligned}$$

(A.5)

and

$$\begin{aligned} \mathrm{E}\big [u\big ((s^{\ell }+s){\tilde{x}}^{\ell }\big )\big ]-\mathrm{E}\big [u\big ((s^{\ell }+s){\tilde{x}}^h\big )\big ] \approx \frac{1}{k!}u^{(k)}\big ((s^{\ell }+s)x^{\circ }\big )\Delta _k(s^{\ell }+s). \end{aligned}$$

(A.6)

Using Eqs. (A.4) and (A.5), we have

$$\begin{aligned} \lim _{s\rightarrow 0} \frac{\pi _p(sx^{\circ },s^{\ell })}{s}\approx & {} \frac{\Delta _k(s^{\ell })}{k!} \lim _{s\rightarrow 0} \frac{u^{(k)}(s^{\ell }x^{\circ })-u^{(k)}\Big ((s^{\ell }+s)x^{\circ }\Big )}{s} \nonumber \\= & {} \frac{u^{(k)}(s^{\ell }x^{\circ })\Delta _k(s^{\ell })}{k!s^{\ell }}\times -\frac{s^{\ell }}{u^{(k)}(s^{\ell }x^{\circ })} \frac{\partial u^{(k)}\Big ((s^{\ell }+s)x^{\circ }\Big )}{\partial s}\bigg |_{s=0} \nonumber \\= & {} \frac{u^{(k)}(s^{\ell }x^{\circ })\Delta _k(s^{\ell })}{k!s^{\ell }}\times -s^{\ell }x^{\circ } \frac{u^{(k+1)}(s^{\ell }x^{\circ })}{u^{(k)}(s^{\ell }x^{\circ })}. \end{aligned}$$

(A.7)

Using Eqs. (A.5) and (A.6), we have

$$\begin{aligned} \lim _{s\rightarrow 0} \frac{\pi _u(0,s^{\ell }+s)-\pi _u(sx^{\circ },s^{\ell })}{s}\approx & {} \frac{u^{(k)}(s^{\ell }x^{\circ })}{k!} \lim _{s\rightarrow 0} \frac{\Delta _k(s^{\ell }+s)-\Delta _k(s^{\ell })}{s} \nonumber \\= & {} \frac{u^{(k)}(s^{\ell }x^{\circ })\Delta _k(s^{\ell })}{k!s^{\ell }}\times \frac{s^{\ell }}{\Delta _k(s^{\ell })} \frac{\partial \Delta _k(s^{\ell }+s)}{\partial s}\bigg |_{s=0} \nonumber \\= & {} \frac{u^{(k)}(s^{\ell }x^{\circ })\Delta _k(s^{\ell })}{k!s^{\ell }}\times k. \end{aligned}$$

(A.8)

Substituting Eqs. (A.7) and (A.8) into the right-hand side of Eq. (17) yields Eq. (18).