Precautionary saving under many risks

Abstract

This paper studies precautionary saving when many small risks are considered. We first introduce two simultaneous risks: labor income and interest rate risks. We show that, in this context, sufficient conditions for precautionary saving are weaker than in similar models. Moreover, we find that, unlike previous literature, precautionary saving can occur in the case of negative covariance between the two risks and in the case of imprudence. We then extend our analysis to a three-risk framework, where a background risk is included. We derive sufficient conditions for precautionary saving which are interpreted in the light of the previous literature.

Appendix

In this Appendix we consider a modification of Problem 6 where background risk is introduced in both first and second periods. The framework is the same as Problem 6 except for the introduction of the stochastic term $\tilde{x_0}=x_0+\psi$, where $\psi$ is a random variable such that $\mathbb E [\psi ] =0$ and $\mathbb E [ \tilde{x_0}] =x_0$. The consumer decision problem thus becomes:

$$\begin{aligned} \max _{s}\mathbb E [u( y_{0}-s,\tilde{x_0})]+\mathbb E [ v(\tilde{y}+s(1+\tilde{r}), \tilde{x})]. \end{aligned}$$

(34)

The optimal level of saving $\check{ s}$ is thus determined by the first-order condition:

$$\begin{aligned} \mathbb E [(1+\tilde{r})v_{1}(\tilde{y}+\check{ s}(1+\tilde{r}),\tilde{x})]-\mathbb E [u_{1}(y_{0}-\check{ s},\tilde{x_0})]=0 \end{aligned}$$

(35)

The condition which ensures positive precautionary saving is obtained by comparing Eqs. (7) and (35). It is clear that, since $u_{11}<0$ and $v_{11}<0$, $\check{s}\ge s^{\star }$ holds if and only if

$$\begin{aligned}&\mathbb E [(1+\tilde{r})v_{1}(\tilde{y}+\check{ s}(1+\tilde{r}),\tilde{x})]-(1+r)v_{1}(y+\check{ s}(1+r),x_1)\nonumber \\&\quad +\,u_{1}(y_{0}-\check{ s},x_0)-\mathbb E [u_{1}(y_{0}-\check{ s},\tilde{x_0})]\ge 0 \end{aligned}$$

(36)

Computations similar to those in Sect. 4 yield:

Corollary 12

In the case of small labor income and interest rate risks in the second period ($\tilde{y}$ and $\tilde{r}$, respectively) and small background risks in both first and second periods ($\tilde{x_0}$ and $\tilde{x}$, respectively) and assuming positive covariances between the random variables $\tilde{y}$, $\tilde{r}$ and $\tilde{x}$, the five Conditions (28), (29), (30) and (31) and

$$\begin{aligned} u_{122}\le 0 \end{aligned}$$

(37)

are sufficient to have positive precautionary saving ($\check{ s}\ge s^\star$).

Proof

As in Corollary 11, Conditions (28), (29), (30) and (31) ensure that the difference between the first and the second terms in the left-hand side of Inequality (36) is positive. By Jensen’s inequality, Condition (37) ensures that the difference between the third and the fourth terms in the left-hand side of Inequality (36) is positive too. All the conditions together thus ensure that Inequality (36) holds. $\square$

The first four conditions in Corollary 12 are the same as Corollary 11 and their interpretation is analogous to that in Sect. 4. Condition (37), on the other hand, is new and is directly related to the introduction of a background risk in the first period. As Condition (29) indicates cross-prudence in wealth in period 2, Condition (37) indicates cross-imprudence in wealth in period 1. Its interpretation is thus straightforward. In presence of a background risk in the second period, the agent is pushed to increase saving if her disutility due to the background risk is lower when second-period wealth is higher. This is what cross-prudence in wealth in second period guarantees. Similarly, when we also have a background risk in first period, this risk pushes the agent to increase saving (and thus to reduce wealth in the first period) if the disutility due to the background risk is lower when first-period wealth is lower. This is exactly what is ensured by cross-imprudence in first period.