On Decreasing Marginal Impatience

Abstract

One of the most controversial assumptions in endogenous time preference theory is that the degree of impatience is marginally increasing in wealth. We examine the implications of an empirically more relevant specification whereby time preference exhibits decreasing marginal impatience (DMI). With DMI, there are multiple steady-state non-satiated and satiated equilibria. In a constant interest rate economy, the non-satiated steady-state point is necessarily unstable. In a capital economy with decreasing returns technology, both the non-satiated and satiated steady-state points can be saddlepoint stable. The model is used to examine policy implications for the effects of capital taxation and government spending.

The original article first appeared in the Japanese Economic Review 59, pages 259–274, 2008. A newly written addendum has been added to this book chapter.

Appendices

Appendix

Stability of Points E ^∗ and E ^∗∗ in Proposition 1

Local optimal dynamics around the non-satiated steady-state point $\left (c^{{\ast}},\phi ^{{\ast}},a^{{\ast}}\right )$ is linearized as:

$$\displaystyle{ \left (\begin{array}{l} \dot{c}\\ \dot{\phi }\\ \dot{a} \end{array} \right ) = \left (\begin{array}{lll} 0 &\frac{r\delta _{c}} {\xi _{\mathit{cc}}} & 0 \\ -\xi _{c} &r &0 \\ - 1&0 &r \end{array} \right )\left (\begin{array}{l} c - c^{{\ast}} \\ \phi -\phi ^{{\ast}} \\ a - a^{{\ast}} \end{array} \right ), }$$

where the coefficient matrix is evaluated at $\left (c^{{\ast}},\phi ^{{\ast}},a^{{\ast}}\right )$. This system has three positive characteristic roots:

$$\displaystyle{ \frac{1} {2}\left \{r + \left (r^{2} -\frac{4r\delta _{c}\xi _{c}} {\xi _{\mathit{cc}}} \right )^{1/2}\right \},\text{ }\frac{1} {2}\left \{r -\left (r^{2} -\frac{4r\delta _{c}\xi _{c}} {\xi _{\mathit{cc}}} \right )^{1/2}\right \},\ \ \text{and}\ \ r, }$$

implying that the non-satiated steady-state point $\left (c^{{\ast}},\phi ^{{\ast}},a^{{\ast}}\right )$ is unstable.

Local optimal dynamics around the satiated steady-state point $\left (c^{{\ast}{\ast}},\phi ^{{\ast}{\ast}},a^{{\ast}{\ast}}\right )$ is linearized as:

$$\displaystyle{ \left (\begin{array}{l} \dot{c}\\ \dot{\phi }\\ \dot{a} \end{array} \right ) = \left (\begin{array}{lll} \delta - r &\frac{r\delta _{c}} {\xi _{\mathit{cc}}} & 0 \\ 0 &\delta &0\\ - 1 &0 &r \end{array} \right )\left (\begin{array}{l} c - c^{{\ast}{\ast}} \\ \phi -\phi ^{{\ast}{\ast}} \\ a - a^{{\ast}{\ast}} \end{array} \right ), }$$

where the coefficient matrix is evaluated at $\left (c^{{\ast}{\ast}},\phi ^{{\ast}{\ast}},a^{{\ast}{\ast}}\right )$. This system has characteristic roots r, δ(c ^∗∗), and δ(c ^∗∗) − r, which is negative as $\delta (c^{{\ast}{\ast}}) <\delta (c^{{\ast}}) = r$ from Assumption 6. The satiated steady-state point $\left (c^{{\ast}{\ast}},\phi ^{{\ast}{\ast}},a^{{\ast}{\ast}}\right )$ is thus saddlepoint stable.

Proof of Lemma 1

By linearizing system in Eq. (11.19) around the non-satiated steady-state point $\left (c^{{\ast}},\phi ^{{\ast}},k^{{\ast}}\right )$, the local dynamic system can be obtained as:

$$\displaystyle{ \left (\begin{array}{l} \dot{c}\\ \dot{\phi }\\ \dot{k} \end{array} \right ) = \left (\begin{array}{lll} 0 & \frac{\delta \delta _{c}} {\xi _{\mathit{cc}}} & -\frac{\xi _{c}\left (1-\tau \right )f_{\mathit{kk}}} {\xi _{\mathit{cc}}} \\ -\xi _{c} &\delta &0 \\ - 1&0 &f_{k} \end{array} \right )\left (\begin{array}{l} c - c^{{\ast}} \\ \phi -\phi ^{{\ast}} \\ a - a^{{\ast}} \end{array} \right ), }$$

where the coefficient matrix is evaluated at $\left (c^{{\ast}},\phi ^{{\ast}},k^{{\ast}}\right )$. For this coefficient matrix:

$$\displaystyle{ \text{trace} =\delta +f_{k} > 0,\text{ det. } = \frac{\delta \xi _{c}} {\xi _{cc}}\left (\delta _{c}f_{k} -\left (1-\tau \right )f_{\mathit{kk}}\right ). }$$

The linear system thus has one negative and two positive roots if and only if $\delta _{c}\left (c^{{\ast}}\right )f_{k}\left (k^{{\ast}}\right ) -\left (1-\tau \right )f_{\mathit{kk}}\left (k^{{\ast}}\right ) > 0$, as stated as the first item in Lemma 1.

By linearizing the system equation (11.19) around the satiated steady-state point $\left (c^{{\ast}{\ast}},\phi ^{{\ast}{\ast}},k^{{\ast}{\ast}}\right )$, the local dynamic system can be obtained as:

$$\displaystyle{ \left (\begin{array}{l} \dot{c}\\ \dot{\phi }\\ \dot{k} \end{array} \right ) = \left (\begin{array}{lll} \delta -\left (1-\tau \right )f_{k}& \frac{\delta \delta _{c}} {\xi _{\mathit{cc}}} & 0 \\ 0 &\delta &0\\ - 1 &0 &f_{ k} \end{array} \right )\left (\begin{array}{l} c - c^{{\ast}{\ast}} \\ \phi -\phi ^{{\ast}{\ast}} \\ a - a^{{\ast}{\ast}} \end{array} \right ), }$$

where the coefficient matrix is evaluated at $\left (c^{{\ast}{\ast}},\phi ^{{\ast}{\ast}},k^{{\ast}{\ast}}\right )$. The characteristic roots are f _k , δ, and $\delta -\left (1-\tau \right )f_{k}$. The linear system thus has one negative and two positive roots if and only if $\delta \left (c^{{\ast}{\ast}}\right ) -\left (1-\tau \right )f_{k}\left (k^{{\ast}{\ast}}\right ) < 0$, as stated as the second statement in Lemma 1.

Addendum: Related Studies

In the text article, we formalize DMI by specifying the subjective discount rate as a function of consumption (or instantaneous utility).¹⁴ Alternatively, it is possible to introduce DMI by assuming that the subjective discount rate is a decreasing function with respect to wealth (e.g., Schumacher 2009) or saving (e.g., Gootzeit et al. 2002). Becker and Mulligan (1997) deal with DMI in a “future-oriented capital” model, in which accumulation of the future-oriented capital leads to a lower discount rate, so that wealthier people become more patient. In either case, similar economic implications of DMI, as discussed in our article, could be obtained.

Literature on dynamic macroeconomic theory incorporates DMI for various purposes, such as to analyze growth dynamics in an overlapping generations model (e.g., Sarkar 2007) and in an AK model with borrowing constraints (e.g., Borissov 2013). It has helped investigate asset pricing in an overlapping generations model (e.g., Nath and Sarkar 2006) and to examine equilibrium indeterminacy in response to interest-rate rules (e.g., Chang et al. 2011). It also allows us to consider the effects of inflation on capital accumulation (e.g., Hirose and Ikeda 2004b; Gong 2006; Chen et al. 2008).

Based on the text article, we have been advancing further research on DMI. Hirose and Ikeda (2012a,b) investigate implications of DMI in a two-country world economy. If both countries exhibit DMI, the steady-state equilibrium is always unstable. For saddle-point stability, at least one country needs to exhibit IMI. Hirose and Ikeda (2012a) analyze the equilibrium dynamics in a one-good, two-country model where one country has DMI and the other has IMI.

Hirose and Ikeda (2012b) solve for two-good, two-country equilibrium dynamics with endogenous time preference, and re-examine the Harberger-Laursen-Metzler (hereafter HLM) effect, which states that a terms-of-trade deterioration would cause a reduction in national savings and a current-account deficit. Although the HLM effect is invalid for a small country with IMI preference (as shown in Obstfeld 1982), it can be rehabilitated in a two-country economy. The terms-of-trade deterioration affects the long-run accumulation of net foreign assets and hence the current account through the following three channels: (a) the income-compensating effect (which is always positive), (b) the welfare-supporting effect, and (c) the interest-income effect. In the case where both countries have IMI, the HLM effect can materialize if the negative welfare-supporting effect dominates the positive income-compensating effect. In the case where one country exhibits DMI and the other exhibits IMI, the HLM effect is necessarily invalid for the IMI country (since the welfare-supporting effect is positive) whereas it may be valid for the DMI country (due to the negative welfare-supporting effect).

As for empirical research, it is a matter of controversy as to how time preference and the discount rate relate to the decision maker’s degree of affluence, measured by income and/or wealth. However, the majority of previous research reports that the degree of impatience, measured by time preference or personal discount rate, is negatively associated with income and/or wealth. Table 11.1 summarizes the previous literature. Although it covers only a part of the literature, 12 of the 17 studies listed indicate that richer people are more patient, as per the DMI model.

Table 11.1 Associations between time preference (discounting) and income

Note, however, that the detected associations do not capture any causality. In particular, since more patient people would have higher saving propensity and hence be wealthier, there could be an endogeneity problem when estimating how time preference relates to income and wealth. The previous studies in Table 11.1 do not cope with the problem.¹⁵ It is an important research topic to tackle this problem and thereby detect causal relationship between time preference and income/wealth.