Hand-to-mouth consumers, rule-of-thumb savers, and optimal control

Abstract

Contemporaneous research in macroeconomics is experiencing a methodological transition. The well-established representative agent optimal control model, which has served for many decades as the fundamental point of reference to approach most of the aggregate phenomena in economics, is being gradually adapted or replaced. In the new settings, agent heterogeneity prevails: while some agents eventually continue to take optimal intertemporal decisions, others do not possess the ability or the willingness to do so, and therefore resort to simple heuristics in their decision-making processes. In this study, the basic Ramsey growth model of intertemporal choice is reinterpreted in the light of the heterogeneity assumption. Four different frameworks are proposed, and the respective dynamics investigated; these frameworks contemplate: (i) the coexistence of optimal planners and hand-to-mouth consumers; (ii) the coexistence of optimal planners and rule-of-thumb savers; (iii) intertemporal discount rate heterogeneity with endogenous determination of hand-to-mouth behavior; (iv) absence of optimal planning and full heuristic behavior. The derived results point to a richness of outcomes that can only be unveiled once the simple dynamic growth setup is expanded to include different consumption-savings profiles.

Appendix: Proofs of Propositions

Proof of Proposition 1

The steady-state consumption of an optimal planner is displayed in Eq. (10); the consumption of a hand-to-mouth decision-maker is $\left( c^{H}\right) ^{*}=(1-\alpha )f[(1-\lambda )\left( k^{O}\right) ^{*}]$ . Condition $\left( c^{O}\right) ^{*}>\left( c^{H}\right) ^{*}$ is equivalent to $\left( \frac{1}{\alpha }\rho +\frac{1-\alpha }{\alpha }\delta \right) \left( k^{O}\right) ^{*}>(1-\alpha )f[(1-\lambda )\left( k^{O}\right) ^{*}]$. Replacing, in this inequality, ratio $\alpha \frac{ f[(1-\lambda )\left( k^{O}\right) ^{*}]}{\left( k^{O}\right) ^{*}}$ by the equivalent expression in Eq.  (9), it is straightforward to simplify the inequality to $\rho +(1-\alpha )\lambda \delta >0$. This is a true condition, implying that, independently of the value of parameter $\lambda$, consumption of a planner is always a higher value, in the steady-state, than the consumption of a hand-to-mouth agent.

Proof of Proposition 2

Solve $\left( c^{O}\right) ^{*}>\left( c^{H}\right) ^{*},$ with the value of $\left( c^{O}\right) ^{*}$ given by (14) and the value of $\left( c^{H}\right) ^{*}$ equal to the marginal productivity of labor, i.e., $\frac{1}{1+\tau }\left( \frac{1}{\alpha }\rho +\frac{1-\alpha }{\alpha }\delta \right) \left( k^{O}\right) ^{*}>(1-\alpha )f[(1-\lambda )\left( k^{O}\right) ^{*}]$. Resorting to Eq. (9) to simplify the inequality and rearranging the corresponding expression, one obtains condition (15).

Proof of Proposition 4

Taking into consideration expressions (29) and (31), variable $c^{S}(t)$ is, in the steady-state, equivalent to:

$$\begin{aligned} \left( c^{S}\right) ^{*}=\frac{1-s}{s}\delta k^{*} \end{aligned}$$

(49)

The steady-state consumption of an optimal planner is calculated from $\overset{\text {{.}}}{k^{O}}(t)=0$, given the already computed steady-state relations,

$$\begin{aligned} \left( c^{O}\right) ^{*}=\left[ \frac{\left( 1+\frac{\alpha }{1-{\widehat{\lambda }}}\Psi \right) \left( 1-\frac{1}{{\widehat{\lambda }}}\Psi \right) }{ s\left( 1-\frac{\alpha }{{\widehat{\lambda }}}\Psi \right) }-\left( 1+\frac{1}{ 1-{\widehat{\lambda }}}\Psi \right) \right] \delta k^{*} \end{aligned}$$

(50)

If $s=\frac{\alpha \delta }{\rho +\delta }$ and, hence, $\Psi =0$, it is straightforward to arrive to outcome $\left( c^{S}\right) ^{*}=\left( c^{O}\right) ^{*}=\left( \frac{1}{\alpha }\rho +\frac{1-\alpha }{\alpha } \delta \right) k^{*}$. In the scenario where $s<\frac{\alpha \delta }{ \rho +\delta }$, one knows that $-\left( 1-{\widehat{\lambda }}\right)<\Psi <0$. Replacing any value of $\Psi$ within this interval into expression ( 50), one obtains a value of $\left( c^{O}\right) ^{*}$ larger than $\frac{1-s}{s}\delta k^{*}$, i.e., larger than $\left( c^{S}\right) ^{*}$. The opposite occurs for $s>\frac{\alpha \delta }{\rho +\delta }$; in this case $0<\Psi <{\widehat{\lambda }}$, and, for any $\Psi$ such that $\left( c^{O}\right) ^{*}>0$, $\left( c^{O}\right) ^{*}$ is maintained in a value that is lower than $\frac{1-s}{s}\delta k^{*}$.

Proof of Proposition 5

Hand-to-mouth consumers will consume their wage income, i.e.,

$$\begin{aligned} \left( c^{H}\right) ^{*}=(1-\alpha )f(k^{*}) \end{aligned}$$

(51)

while the steady-state consumption level of an optimizer is calculated by applying condition $\overset{\text {{.}}}{k}(t,i)=0$ to constraint (36),

$$\begin{aligned} c^{*}(i)=\frac{1}{1-\alpha }\left[ \frac{f(k^{*})}{k^{*}}+\frac{ \rho (i)+\delta }{\alpha }\delta \frac{k^{*}}{f(k^{*})}-\rho (i)-(3-\alpha )\delta \right] k^{*} \end{aligned}$$

(52)

Whether $c^{*}(i)$ increases or falls with a change in $\rho (i)$ depends on the sign of the following partial derivative of (52): $\frac{\partial c^{*}(i)}{\partial \rho (i)}=\frac{1}{1-\alpha }\left[ \frac{\delta }{\alpha }\frac{k^{*}}{f(k^{*})}-1\right] k^{*}$. If $\frac{f(k^{*})}{k^{*}}>\frac{\delta }{\alpha }$, then $\frac{ \partial c^{*}(i)}{\partial \rho (i)}<0$ and $c^{*}(i)$ decreases as the discount rate $\rho (i)$ increases; the opposite occurs for $\frac{ f(k^{*})}{k^{*}}<\frac{\delta }{\alpha }$. In the boundary case $\frac{f(k^{*})}{k^{*}}=\frac{\delta }{\alpha }$, $c^{*}(i)=\frac{ 1-\alpha }{\alpha }\delta k^{*}$ for all feasible values of $\rho (i)$. Note, as well, that if $\frac{f(k^{*})}{k^{*}}=\frac{\delta }{\alpha }$, then, according to (51), also $\left( c^{H}\right) ^{*}= \frac{1-\alpha }{\alpha }\delta k^{*}$.

Regardless of the upward or downward movement of $c^{*}(i)$ as the discount rate increases, the threshold point that marks the transition from planning to hand-to-mouth behavior, which takes place at the value of $\rho (i)$ that satisfies condition $\frac{f(k^{*})}{k^{*}}=\frac{\rho (i)+\delta }{(2-\alpha )\alpha }$, is always such that $c^{*}(i)=\left( c^{H}\right) ^{*}$. Hence, three outcomes are possible, depending on the value of ratio $\frac{f(k^{*})}{k^{*}}$. If $\frac{f(k^{*})}{ k^{*}}=\frac{\delta }{\alpha }$, then $c^{*}(i)=\left( c^{H}\right) ^{*}=\frac{1-\alpha }{\alpha }\delta k^{*},\forall$ $\rho (i)\ge 0$ ; if $\frac{f(k^{*})}{k^{*}}>\frac{\delta }{\alpha }$, then $c^{*}(i)$ starts at a value such that $c^{*}(i)>\left( c^{H}\right) ^{*}$ for $\rho (i)=0$ and it declines until it reaches $c^{*}(i)=\left( c^{H}\right) ^{*}$ at point $\frac{f(k^{*})}{k^{*}}=\frac{\rho (i)+\delta }{(2-\alpha )\alpha }$; if $\frac{f(k^{*})}{k^{*}}<\frac{ \delta }{\alpha }$, then $c^{*}(i)$ starts at a value such that $c^{*}(i)<\left( c^{H}\right) ^{*}$ for $\rho (i)=0$ and it increases until it reaches $c^{*}(i)=\left( c^{H}\right) ^{*}$ at point $\frac{ f(k^{*})}{k^{*}}=\frac{\rho (i)+\delta }{(2-\alpha )\alpha }$.

Proof of Proposition 6

The first two conditions of dynamic equation (46) imply that k(t, i) converges to zero over time: agents will not accumulate capital in the long-term and their consumption level is $\left( c^{H}\right) ^{*}=w^{*}=(1-\alpha )f(k^{*})$, i.e., these agents are hand-to-mouth consumers. The remaining agents are rule-of-thumb savers for which we derive the following steady-state outcome,

$$\begin{aligned} -(1-\zeta )\sigma (i)\left[ (1-\alpha )+\alpha \frac{k^{*}(i)}{k^{*}} \right] f\left( k^{*}\right) =\delta k^{*}(i) \end{aligned}$$

(53)

From consumption function (44), it is straightforward to compute the equality in the proposition, once one takes into account relation (53).