Indeterminacy in stochastic overlapping generations models: real effects in the long run

Zhigang Feng; Matthew Hoelle

doi:10.1007/s00199-015-0947-y

Economic Theory

February 2017, Volume 63, Issue 2, pp 559–585 | Cite as

Indeterminacy in stochastic overlapping generations models: real effects in the long run

Authors
Authors and affiliations

Zhigang Feng
Matthew Hoelle

Research Article

First Online: 26 December 2015

Received: 25 March 2015

Accepted: 13 December 2015

Abstract

Indeterminate equilibria are known to exist for overlapping generations models, though recent research has been limited to deterministic settings in which all equilibria converge to a steady state in the long run. This paper analyzes stochastic overlapping generations models with three-period-lived representative consumers and adopts a novel computational algorithm to numerically approximate the entire set of competitive equilibria. In a stochastic setting with incomplete markets, indeterminacy has real effects in the long run. Our numerical simulations reveal that indeterminacy is an order of magnitude more important than endowment shocks in explaining long-run consumption and asset price volatility.

Keywords

OLG Indeterminacy Markov Equilibrium Computation Simulation

JEL Classification

C63 D52 D91 E21

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Appendix

Proof of Theorem 1

To show that a Markov equilibrium satisfies the SCE definition, the Euler Eqs. (9) and (11) must be necessary and sufficient for household optimality. Necessity is immediate. Sufficiency follows as households are finite-lived.

Proof of Theorem 2

From the standard arguments used to show the existence of a SCE, the set of shadow prices of investment $m\left( s^{t}\right) $ belong to a compact set $\Delta \subseteq \mathbb {R}^{J}$ in all nodes. The iterative construction begins with the correspondence $\mathbf {V}_{0}:{\mathbf {S}\times \mathbb {R} }\rightrightarrows {\mathbb {R}}_{+}^{J}$ such that $\mathbf {V} _{0}\left( s,\omega \right) =\Delta $ $\forall \left( s,\omega \right) \in {\mathbf {S}\times \mathbb {R}.}$

Given a correspondence $\mathbf {V}_{n}:{\mathbf {S}\times \mathbb {R} }\rightrightarrows {\mathbb {R}}_{+}^{J}$ for any $n\ge 0$, define an operator $\mathbf {B}$ that maps the correspondence $\mathbf {V}_{n} :{\mathbf {S}\times \mathbb {R}}\rightrightarrows {\mathbb {R}}_{+}^{J}$ to a new correspondence $\mathbf {V}_{n+1}:{\mathbf {S}\times \mathbb {R}} \rightrightarrows {\mathbb {R}}_{+}^{J}$ defined as follows:

$$\begin{aligned}&\mathbf {V}_{n+1}\left( s,\omega \right) \\&\quad =\left\{ m\in \Delta : \begin{array}{l} \hbox {for }\left( q,\theta _{1}\right) =\mathbf {f}\left( s,\omega ,m\right) ,\\ \hbox {there exists }m^{\prime }\left( \sigma \right) \in \mathbf {V}_{n}\left( \sigma ,r(\sigma )\theta _{1}\right) \qquad \forall \sigma \in \mathbf {S}\quad \hbox {and}\\ \left( q^{\prime }\left( \sigma \right) ,\theta _{1}^{\prime }\left( \sigma \right) \right) =\mathbf {f}\left( \sigma ,r(\sigma )\theta _{1},m^{\prime }\left( \sigma \right) \right) \qquad \forall \sigma \in \mathbf {S} \quad \hbox {such that}\\ q^{j}u_{c}\left[ \mathbf {e}_{0}(s)+q\theta _{1}\right] =\beta \sum \limits _{\sigma \in \mathbf {S}}\pi (s,\sigma )\frac{m^{\prime j}\left( \sigma \right) }{q^{\prime j}\left( \sigma \right) }r^{j}\left( \sigma \right) \quad \forall j\in \mathbf {J} \end{array} \right\} . \end{aligned}$$

The correspondences are defined recursively using this operator $\mathbf {B}:$

$$\begin{aligned} \mathbf {V}_{n+1}=\mathbf {B}\left( \mathbf {V}_{n}\right) . \end{aligned}$$

The Markov equilibrium policy correspondence is defined as follows:

$$\begin{aligned} \mathbf {V}^{*}\left( s,\omega \right) =\lim _{n\rightarrow \infty } \mathbf {B}\left( \mathbf {V}_{n}\left( s,\omega \right) \right) \quad \forall \left( s,\omega \right) \in {\mathbf {S}\times \mathbb {R}.} \end{aligned}$$

Theorem 1 from Feng et al. (2014), reproduced below, guarantees the existence of a Markov equilibrium policy correspondence.

Theorem 6

Let $\mathbf {V}_{0}$ be a compact-valued correspondence such that $\mathbf {V}_{0}\supset \mathbf {V}^{*}$. Let $\mathbf {V}_{n+1}=\mathbf {B} \left( \mathbf {V}_{n}\right) ,n\ge 0$. Then, $\mathbf {V}_{n}\rightarrow \mathbf {V}^{*}$ as $n\rightarrow \infty $. Moreover, $\mathbf {V} ^{*}$ is the largest fixed point of the operator $\mathbf {B}$, i.e., if $\mathbf {V}=\mathbf {B(V)}$, then ${\mathbf {V}\subset \mathbf {V}}^{*}$.

Proof of Theorem 4

Suppose, in order to obtain a contradiction, that the markets are complete. This implies that the allocation is (interim) Pareto efficient. The equilibrium allocation is stationary (Henriksen and Spear 2012). This implies that both $\mathbf {V}^{*}$ and $\mathbf {F}$ are single-valued correspondences.

Proof of Theorem 5

The transition correspondence $\mathbf {F}:graph(\mathbf {V}^{*})\rightrightarrows \left( {\mathbb {R}\times \mathbb {R}}_{+}^{J}\right) ^{S}$ defined by

$$\begin{aligned}&\mathbf {F}\left( s,\omega ,m\right) \\&\quad \! =\left\{ \left( \omega ^{\prime } (\sigma ),m^{\prime }\left( \sigma \right) \right) _{\sigma \in \mathbf {S}}: \begin{array}{c} \omega ^{\prime }(\sigma )=r(\sigma )\mathbf {f}_{\theta }\left( s,\omega ,m\right) \\ m^{\prime }\left( \sigma \right) \in \mathbf {V}^{*}\left( \sigma ,\omega ^{\prime }(\sigma )\right) \\ q^{\prime }\left( \sigma \right) =\mathbf {f}_{q}\left( \sigma ,\omega ^{\prime }(\sigma ),m^{\prime }\left( \sigma \right) \right) \\ q^{j}u_{c}\left[ \mathbf {e}_{0}(s)+q\theta _{1}\right] =\beta \sum \limits _{\sigma \in \mathbf {S}}\pi (s,\sigma )\frac{m^{\prime k}\left( \sigma \right) }{q^{\prime k}\left( \sigma \right) }r^{j}\left( \sigma \right) \quad \forall j,k\in \mathbf {J} \end{array} \right\} . \end{aligned}$$

By definition of $\mathbf {f}_{q}\left( \sigma ,\omega ^{\prime }(\sigma ),m^{\prime }\left( \sigma \right) \right) :$

$$\begin{aligned} \frac{m^{\prime k}\left( \sigma \right) }{q^{\prime k}\left( \sigma \right) }=\frac{m^{\prime 1}\left( \sigma \right) }{q^{\prime 1}\left( \sigma \right) }\quad \forall k. \end{aligned}$$

This means that the J Euler equations given by:

$$\begin{aligned} q^{j}u_{c}\left[ \mathbf {e}_{0}(s)+q\theta _{1}\right] =\beta \sum \limits _{\sigma \in \mathbf {S}}\pi (s,\sigma )\frac{m^{\prime 1}\left( \sigma \right) }{q^{\prime 1}\left( \sigma \right) }r^{j}\left( \sigma \right) \quad \forall j\in \mathbf {J.} \end{aligned}$$

Suppose, in order to obtain a contradiction, that incomplete markets indeterminacy does not hold. This implies that $\mathbf {F}\left( s,\omega ,m\right) $ is determinate (0-dimensional image) for all state variables $\left( s,\omega ,m\right) $. Under initial condition indeterminacy, $\mathbf {V}^{*}\left( \sigma ,\omega ^{\prime }(\sigma )\right) $ is indeterminate (strictly positive dimension) for all state variables $\left( s,\omega ,m\right) $ (with corresponding state variable $\left( \sigma ,\omega ^{\prime }(\sigma )=r(\sigma )\mathbf {f}_{\theta }\left( s,\omega ,m\right) \right) $). This implies that for all state variables $\left( s,\omega ,m\right) $, $\exists !\left( \omega ^{\prime }(\sigma ),m^{\prime }\left( \sigma \right) \right) _{\sigma \in \mathbf {S}}$ such that the Euler equations are satisfied.¹⁴ If $\left( \omega ^{\prime }(\sigma ),m^{\prime }\left( \sigma \right) \right) _{\sigma \in \mathbf {S}}$ is uniquely determined, then with $q^{\prime }\left( \sigma \right) =\mathbf {f}_{q}\left( \sigma ,\omega ^{\prime }(\sigma ),m^{\prime }\left( \sigma \right) \right) $, the vector $\left( \frac{m^{\prime 1}\left( \sigma \right) }{q^{\prime 1}\left( \sigma \right) }\right) _{\sigma \in \mathbf {S}}$ is uniquely determined.

If $\left( \frac{m^{\prime 1}\left( \sigma \right) }{q^{\prime 1}\left( \sigma \right) }\right) _{\sigma \in \mathbf {S}}$ is uniquely determined, the Euler equations for both the young and middle-aged consumers imply:

$$\begin{aligned} \frac{\left( u_{c}\left[ \mathbf {e}_{1}(\sigma )-\omega ^{\prime }\left( \sigma \right) -q^{\prime }\left( \sigma \right) \theta _{1}^{\prime }\left( \sigma \right) \right] \right) _{\sigma \in \mathbf {S}}}{u_{c}\left[ \mathbf {e}_{0}(s)+q\theta _{1}\right] }=\frac{\left( u_{c}\left[ \mathbf {e}_{2}(\sigma )+\omega ^{\prime }\left( \sigma \right) \right] \right) _{\sigma \in \mathbf {S}}}{u_{c}\left[ \mathbf {e}_{1}(s)-\omega -q\theta _{1}\right] }. \end{aligned}$$

For all state variables $\left( s,\omega ,m\right) $, it is not possible to find a Pareto-improving reallocation. This means that the allocation is (interim) Pareto efficient, meaning that markets are complete. This completes the argument.

Numerical algorithm

The vector of possible values for bond-holding and shocks are given by $\hat{\varvec{\Theta }}=\left\{ \theta _{0}^{i_{1}}\right\} _{i_{1} =1}^{N_{\theta }}$, $\hat{\mathbf {S}}=\left\{ s_{0}^{i_{2}}\right\} _{i_{2}=1}^{N_{s}}$. For each pair of the bond-holding and shock grids, $\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) $, we also define a finite vector of possible values for the image of the correspondence: $\hat{\mathbf {V}}_{0}^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) =\left\{ m_{0}^{i_{1},i_{2},j}\right\} _{j=1}^{Nv}$.¹⁵ Notice, $\lim _{N_{\theta }\rightarrow \infty }\hat{\varvec{\Theta }}=\varvec{\Theta }$, $\lim _{Nv\rightarrow \infty }\hat{\mathbf {V}}_{0}^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) =\tilde{\mathbf {V}}_{0} ^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) $. Finally, we construct the discrete version of operator $\mathbf {B}^{h,\mu ,N}$ by eliminating points (in the Euler equation, for a predetermined tolerance $\epsilon >0$) as follows:

1.

Given $\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) $, pick a point $m_{0}^{i_{1},i_{2},j}$ in the vector $\hat{\mathbf {V}}_{0}^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) $. From $m_{0}^{i_{1} ,i_{2},j}$, we can determine the values of $\left( \theta ^{^{\prime } i_{1},i_{2},j},q^{i_{1},i_{2},j}\right) $ by solving for
$$\begin{aligned} m_{0}^{i_{1},i_{2},j}-q^{i_{1},i_{2},j}\cdot u_{c}\left( e_{1}(s_{0}^{i_{2} })+\theta _{0}^{i_{1}}-q^{i_{1},i_{2},j}\theta ^{^{\prime }i_{1},i_{2},j}\right)&=0, \end{aligned}$$

(12)

$$\begin{aligned} m_{0}^{i_{1},i_{2},j}-\beta \sum _{s^{\prime }}\pi (s^{\prime }|s_{0})u_{c}\left( e_{2}(s^{\prime ^{\prime }i_{1},i_{2},j}\right)&=0. \end{aligned}$$

(13)
Thus, if for all $m^{\prime }\in \hat{\mathbf {V}}_{0}^{\mu ,\varepsilon } (\theta ^{^{\prime }i_{1},i_{2},j},s^{\prime })=\left\{ m^{^{\prime }l} (\theta ^{^{\prime }i_{1},i_{2},j},s^{\prime })\right\} _{l=1}^{N_{V}}$ we have
$$\begin{aligned} \min _{m^{\prime }\in \left\{ m^{\prime l}\right\} _{l=1}^{N_{V}}}\left\| q^{i_{1},i_{2},j}\cdot u_{c}\left( e_{0}(s_{0}^{i_{2}})-q^{i_{1},i_{2} ,j}\theta ^{^{\prime }i_{1},i_{2},j}\right) \!-\!\beta \sum \pi (s^{\prime } |s_{0}^{i_{2}})\left( \frac{m^{\prime }}{q^{\prime }}\right) \right\| >\epsilon \nonumber \\ \end{aligned}$$

(14)
where the value of $q^{\prime }$ is determined by the same procedure in finding $\big ( \theta ^{^{\prime }i_{1},i_{2},j},q^{i_{1},i_{2},j}\big ) $, then $\hat{\mathbf {V}}_{1}^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0} ^{i_{2}}\right) =\hat{\mathbf {V}}_{0}^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0}^{i_{2}}\right) -m_{0}^{i_{1},i_{2},j}$.
2.

Iterate over all possible values $m_{0}^{i_{1},i_{2},j}\in \hat{\mathbf {V}}_{0}^{\mu ,\varepsilon }\left( \theta _{0}^{i_{1}},s_{0}^{i_{2} }\right) $, and all possible $\left( \theta _{0}^{i_{1}},s_{0}^{i_{2} }\right) \in \hat{\varvec{\Theta }}\times \hat{\mathbf {S}}$.
3.

Iterate until convergence is achieved $\sup \left\| \hat{\mathbf {V} }_{n}^{\mu ,\varepsilon }-\hat{\mathbf {V}}_{n-1}^{\mu ,\varepsilon }\right\| =0$.

At the limit of the above algorithm, we have $\lim _{n\rightarrow \infty }\hat{\mathbf {V}}_{n}^{\mu ,\varepsilon }=\hat{\mathbf {V}}^{\mu ,\varepsilon ^{*}}$ (Fig. 1).

Fig. 1
Equilibrium set of $\left\{ \left\{ m^{\prime }(\sigma )\right\} _{\sigma \in \{1,2\}},m\right\} $ at given $\{s,\omega \}$

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Zhigang Feng
- 1
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Matthew Hoelle
- 2

1.Department of EconomicsUniversity of Illinois at Urbana–ChampaignUrbanaUSA
2.Department of EconomicsPurdue UniversityWest LafayetteUSA

Indeterminacy in stochastic overlapping generations models: real effects in the long run

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