Habit formation and indeterminacy in overlapping generations models

Abstract

I introduce habit formation into an otherwise standard overlapping generations economy with pure exchange populated by three-period-lived agents. Habits are modeled in such a way that current consumption increases the marginal utility of future consumption. With logarithmic utility functions, I demonstrate that habit formation gives rise to stable monetary steady states in economies with hump-shaped endowment profiles and reasonably high discount factors. Intuitively, habits imply adjacent complementarity in consumption, which in turn explains why income effects are sufficiently strong in spite of the logarithmic utility. The three-period horizon further strengthens the income effect.

Appendix

1.1 Proofs

Proof

(Proposition 1) Note that $a_{1,t}\left[ R_{t+1},R_{t}\right] =\omega _{1}-c_{1,t}$, and $a_{2,t}\left[ R_{t},R_{t-1}\right] =R_{t-1}\omega _{1}+\omega _{2}-\left( R_{t-1}c_{1,t-1}+c_{2,t}\right)$, where $c_{1,t}$ and $c_{2,t+1}$ are defined as (3) and (4), respectively. The market-clearing condition $A_{t+1}\left[R_{t+2},R_{t+1},R_{t}\right]=R_{t}A_{t}\left[R_{t+1},R_{t},R_{t-1}\right]$ can be written as follows:

$$\begin{aligned} R_{t+2}=\frac{\gamma ^2\left( \omega _{1}-R_{t}A_{t}\left[ R_{t+1},R_{t},R_{t-1}\right] \right) }{\delta \left(R_{t+1}\omega _{1}+\omega _{2}\right) -\left( \omega _{1}-R_{t}A_{t}\left[ R_{t+1},R_{t},R_{t-1}\right] \right) \left( R_{t+1}+\gamma \right)} \end{aligned}$$

where $\delta =\left( 1+\beta +\beta ^{2}\right) ^{-1}$. Clearly, this last expression corresponds to (5) in the text. $\square$

Proof

(Proposition 2) At any stationary equilibrium, $R_{t}=R^{*}$. Then, $a_{1} ^{*} =\omega _{1}-c_{1} ^{*}$, where $c_{1} ^{*}=\left( \omega _{1}+\omega _{2}\right) \left\{ \left( 1+\beta +\beta ^{2}\right) \left( 1+\gamma +\gamma ^{2}\right)\right\} ^{-1}$. In the same fashion:

$$\begin{aligned} a_{2} ^{*} =\omega _{1}+\omega _{2} -\{ 1+(1+\beta )\gamma +\beta /\left( 1+\gamma \right) \} c_{1} ^{*} \end{aligned}$$

Hence, $A^{*} \equiv a_{1} ^{*}+a_{2}^{*}$ is nonnegative as long as $\left( 2-\lambda \right)\omega _{1} +\left( 1-\lambda \right)\omega _{2}$ is nonnegative, where $\lambda =\frac{\left( 1+\beta \right) \left( 1+\gamma +\gamma ^{2}\right) +1+2\gamma }{\left( 1+\beta +\beta ^{2}\right) \left( 1+\gamma +\gamma ^{2}\right) \left( 1+\gamma \right) }$. Moreover, $\mathrm sgn (A^{*}) = \mathrm sgn \{\left( 2-\lambda \right)\omega _{1} +\left( 1-\lambda \right)\omega _{2}\}$. $\square$

Proof

(Proposition 5) The aggregate savings function is given by $A_{t} \equiv \alpha _{1,t}+ \alpha _{2,t} = \omega _{1} + \omega _{2} - (c_{1,t} + c_{2,t}) + R_{t-1}(\omega _{1} - c_{1,t-1})$. The partial derivative $\partial A_{t}/ \partial R_{t}$, evaluated at the monetary steady state in which $R_{t-1}=R_{t}=R_{t+1}=R^{*}=1$ (and given that $\omega _{1}+\omega _{2}=1$), is equal to:

$$\begin{aligned} \frac{\partial A_{t}}{\partial R_{t}}=-\left\{ \frac{\partial c_{1,t}}{\partial R_{t}} + \frac{\partial c_{2,t}}{\partial R_{t}} + \frac{\partial c_{1,t-1}}{\partial R_{t}} \right\} \end{aligned}$$

where:

$$\begin{aligned} \frac{\partial c_{1,t}}{\partial R_{t}}&= \frac{ \omega _{1}\{\gamma (1+\gamma )+1\}-1}{(1+\beta + \beta ^2)(1+\gamma + \gamma ^2)^{2}}\\ \frac{\partial c_{2,t}}{\partial R_{t}}&= \frac{\gamma }{(1+\beta + \beta ^2)(1+\gamma +\gamma ^{2})}\left\{ \frac{\beta }{(1+\gamma )^2} + \frac{(1+\beta ) (1+\gamma )\gamma ^{2}+\beta \gamma }{(1+\gamma )(1+\gamma + \gamma ^{2})}\right\} \\ \frac{\partial c_{1,t-1}}{\partial R_{t}}&= \frac{\gamma ^2}{(1+\beta + \beta ^2)(1+\gamma + \gamma ^{2})^{2}} \end{aligned}$$

Provided that $\{\omega _{1}\{\gamma (1+\gamma )+1\}-1\}$ is positive, the savings function $A_{t}$ is undoubtedly decreasing in $R_{t}$ at yields near the golden rule. It is also clear that both $\partial c_{1,t}/ \partial R_{t}$ and $\partial c_{1,t-1}/ \partial R_{t}$ are decreasing functions of $\beta$. These results suggest that if agents got more impatient ($\beta$ decreases), the savings function would become even more negatively sloped. Nevertheless, the derivative $\partial c_{2,t}/ \partial R_{t}$ is a decreasing function of $\beta$ only when $\beta >\overline{\beta }$ (when $\beta$ is large). Otherwise, it is an increasing function of $\beta$. Thus, the behavior of $\partial c_{2,t}/ \partial R_{t}$ as a function of $\beta$ explains why indeterminacy may persist for values of $\beta$ close to 0.5. $\square$

1.2 Derivation of the offer curve

Let $R_{t}=p_{t}/p_{t+1}$. The excess demand function when old is

$$\begin{aligned} \alpha _{t+1}&= c_{2,t+1}-\omega _{2} \\&= \omega _{1}R_{t}-c_{1,t}R_{t} \\&= \omega _{1}R_{t}-\frac{\omega _{1}R_{t}^{2}+\omega _{2}R_{t}}{\left( 1+\beta \right) \left( R_{t}+\gamma \right) } \\&= \frac{R_{t}\left[ \gamma \omega _{1}-\omega _{2}+\beta \omega _{1}\left( R_{t}+\gamma \right) \right] }{\left( 1+\beta \right) \left( R_{t}+\gamma \right) } \end{aligned}$$

where the second and third lines use $\alpha _{t+1}=R_{t}\alpha _{t}$ and $c_{1,t}=\left( \omega _{1}R_{t}+\omega _{2}\right) \left( 1+\beta \right)^{-1}\left( R_{t}+\gamma \right)^{-1}$, respectively. The last expression can be written as follows:

$$\begin{aligned} 0=\lambda _{2}R_{t}^{2}+\lambda _{1}R_{t}+\lambda _{0} \end{aligned}$$

where $\lambda _{2}=\beta \omega _{1}, \lambda _{1}=\left( 1+\beta \right) \gamma \omega _{1}-\omega _{2}-\left( 1+\beta \right) \alpha _{t+1}$, and $\lambda _{0}=-\left( 1+\beta \right) \gamma \alpha _{t+1}$. From this quadratic form, I get:

$$\begin{aligned} \hat{R}=\frac{-\lambda _{1}\pm \left( \lambda _{1}^{2}-4\lambda _{2}\lambda _{0}\right)^{\frac{1}{2}}}{2\lambda _{2}} \end{aligned}$$

By Descarte’s rule of signs, there is only one positive root $\hat{R}^{+}$ (because $\lambda _{2}>0, \lambda _{0}<0$ and $\lambda _{1}\ne 0$). Finally, the level of savings is given by:

$$\begin{aligned} \alpha _{t}=\{\omega _{1}[ \gamma +\beta ( \hat{R}^{+}+\gamma ) ] -\omega _{2}\}( 1+\beta )^{-1} (\hat{R}^{+}+\gamma )^{-1} \end{aligned}$$

The offer curve is the locus of points $(\alpha _{t},\alpha _{t+1})$.