Output volatility and savings in a stochastic Goodwin economy

Abstract

Both the public and the academic discourse of the post-crisis era have produced most controversial views concerning adequate national savings rates. We add to this discussion by analyzing the role of the savings rate for the dynamics of an economy embedded in a turbulent environment. To this end we study the dynamics of a stochastic Goodwin-type business cycle model using a mix of analytical- and simulation techniques. Focussing on a region of the parameter space that exhibits multi-stability, we apply the stochastic sensitivity function technique and Lyapunov exponents to scrutinize the dynamics of the stochastic economic system. We find that the savings rate affects the sensitivity of the economic system, as well as the distribution of economic states. The sensitivity of the system is inversely related to the level of the savings rate. Specifically, we demonstrate that high volatility phases of the cycle vary as the savings rate is changed. For low (high) levels of the savings rate the stochastic Goodwin economy will remain relatively often in sensitive (robust) states. Our theoretical investigation suggests that strategies involving reasonably high national savings rates might help to avoid the negative welfare implications of sensitive and even chaotic income dynamics along the cycle.

Appendix: stochastic sensitivity function method

One of the outstanding features of non-linear dynamic systems is their sensitivity. For instance, sensitivity to initial conditions, i.e. small distortions of initial conditions may lead to substantially different behavior of the system, is intimately linked to chaotic motion. Similarly, sensitive dependence on small stochastic perturbations has been found. Also this type of sensitivity has been linked to chaos in the literature on dynamic systems. Our understanding of non-linear stochastic systems has been furthered by the development of measures of sensitivity of the system’s attractors.

In this appendix, we outline one solution to this problem: the stochastic sensitivity function technique pioneered by Milstein and Ryashko (1995). The approach consists of a mixture of analytical and numerical techniques to measure local sensitivity of an attractor and aggregate the local information into a global measure of sensitivity. The details of the approach are given below for two types of attractors of a continuous time non-linear system: steady states and limit cycles. This focus is motivated by the circumstance that those attractors feature prominently in the economic model discussed in the body of the paper. The approach is neither limited to continuous time systems not to the said type of attractors. Moreover, the SSF technique proves to be versatile in the presence of co-existing attractors.

There is consent concerning the fact that the Kolmogorov-Fokker-Plank equation represents an ideal concept for the probabilistic description of a stochastic dynamic system. Unfortunately the approach turns out to be intractable even in simplest situations. Pragmatic approaches, typically limited to weak noise scenarios based on asymptotics and/or approximation techniques, have proven viable. The SSF technique belongs into this class of methods. It has been devised in the context of the so-called potential methods. Some of the underlying theory has been laid out in Dembo and Zeitouni (2010) and Freidlin and Wentzell (2012).

1.1 The stochastic differential equation system

Consider a system with n dynamic dimensions. The state vector is given by $x(t) \equiv x \in \mathbb {R}^n$ and f(x) is a n-vector function. Together with a constant $\delta \in \mathbb {R}$, a matrix valued function $\sigma (x)$ of dimension $n \times n$ a n-dimensional Wiener process $\omega (t)$ is introduced to model stochastic disturbances. The stochastic differential equation

$$\begin{aligned} \dot{x} = f(x) + \delta \, \sigma (x) \, \dot{\omega } \end{aligned}$$

(6)

serves as the framework for the discussion of the SSF technique. It is assumed that for $\delta =0$ (6) posses at least one steady state $\bar{x}$ and a solution $x = \xi (t)$ of period T associated with a limit cycle l. Each attractor is assumed to be exponentially stable.

1.2 Stochastic steady state(s)

For $\delta > 0$ the stochastic trajectory will traverse the neighborhood of the steady state $\bar{x}$. The associated stochastic attractor can characterized by a stationary probability distribution $\rho (x, \delta )$ which is the solution of the corresponding Fokker–Plank equation. In response to the apparent intractability of that concept pragmatic approaches have been devised for cases in which $\delta$ is small. Asymptotic arguments based on the quasi-potential $v(x) = - \lim _{\delta \rightarrow 0} \ \delta ^2 \log \rho (x, \delta )$ which imply the approximation of the stationary distribution as $\rho (x, \delta ) \approx K \exp \left( - \frac{v(x)}{\delta ^2} \right)$ where $K \in \mathbb {R}_{+}$ serves as a normalizing constant. Implementing this Ansatz Bashkirtseva and Ryashko (2005, 2011) approximate the quasi-potential v(x) by a quadratic form in deviations of the stochastic trajectory from the deterministic steady state $\bar{x}$. The approximate quasi-potential $v(x) \approx \frac{1}{2} (x-\bar{x})' V_{(n,n)} (x-\bar{x})$ is consistent with an asymptotic stationary density $\rho (x, \delta ) \approx K \exp \left( - \frac{(x-\bar{x})' W_{(n,n)}^{-1}(x-\bar{x})}{2 \delta ^2} \right)$ where $W_{(n,n)}$ is the unique solution of equation $FW + WF' = -S$ with $F= \frac{\partial f}{\partial x}(\bar{x})$ and $S=GG'$ with $G=\sigma (\bar{x})$. The symmetric matrix $W_{(n,n)}$ reflects features of the spatial arrangement around $\bar{x}$ and the expected distance of the random trajectory from the steady state. In fact, $\delta ^2 W$ can be interpreted as a covariance matrix. In the two-dimensional case, a confidence ellipse centered at the steady state $(x-\bar{x})' W_{(n,n)}^{-1}(x-\bar{x}) = -2 \, ln(1-p) \, \delta ^2$ where $p \in (0,1)$ can be constructed.

1.3 Stochastic limit cycle

For attractors other than point attractors the aspect of local sensitivity to disturbances plays a role. One should not expect uniform sensitivity to disturbances along the attractor. Ryashko’s effort to construct a measure of sensitivity of a limit cycle to noise proceeds in two steps. First, a hyperplane $\Pi _t$ which is orthogonal to the deterministic cycle is constructed at some point on the cycle.

Bringing in the stochastic cycle L, the intersections of the stochastic trajectory with the hyperplane are determined. In Fig. 11 such intersections are marked as black dots on the $\Pi _t$ line. The dispersion of those intersection points reflects the local sensitivity of the cycle. The variance is considered as a natural measure of dispersion in this context. In the subsequent step, this local sensitivity measure is obtained at points along the cycle, coded in a function $\mu (t)$, and plotted over the time window underlying the cycle. In that sense, the local information concerning sensitivity is aggregated into a global measure of sensitivity.

In the sequel we provide a formal description of the SSF method for stochastic limit cycles. Let $\Pi _t$ denote a hyperplane which is orthogonal to the cycle l at $\xi (t)$ for $t \in [0,T)$. Here, the quasi-potential is approximated by a quadratic form in deviations from the underlying deterministic cycle $v(x) \approx \frac{1}{2} (x-\xi (t))' W^+(t)(x-\xi (t))$ where $W^+$ denotes a generalized inverse. The following approximation of the local stationary probabilistic distribution $\rho _t(x,\delta ) \approx K \exp \left( - \frac{ (x-\xi (t))' W_{(n,n)}^{+} (x-\xi (t))}{2 \delta ^2}\right)$ is consistent with this approximation of the quasi-potential v(x). The stochastic sensitivity matrix W(t) of the cycle L is determined as the unique solution of the Lyapunov equation $\dot{W} = F(t) W + W F(t)' + P(t) S(t) P(t)$ subject to $W(0)=W(T), W(t)r(t) \equiv 0$, where $F(t)= \frac{\partial f}{\partial x}(\xi (t))$, $S(t) = \sigma (\xi (t)) \sigma (\xi (t))'$, $r(t)= f(\xi (t))$, and P(t) denotes the matrix of the projection onto the hyperplane $\Pi _t$.

Fig. 11

Poincaré section(s) $\Pi _t$

In the two-dimensional case, the stochastic sensitivity function reduces to $W(t) = \mu (t) P(t)$ where T-periodic scalar stochastic sensitivity function $\mu (t) > 0$ is a solution to the boundary problem $\dot{\mu } = a(t) \mu + b(t)$ subject to $\mu (0)=\mu (T)$ with the coefficients $a(t) = u(t)' \{ F(t)' + F(t) \} u(t)$ and $b(t)= u(t)'S(t)u(t)$ with u(t) being a normalized vector orthogonal to $f(\xi (t))$. An explicit expression for the solution $\mu (t)$ can be found in Bashkirtseva and Ryashko (2005).

The confidence band around the deterministic cycle is constructed on the basis of $\mu (t)$. For the line $\Pi _t$ orthogonal to the cycle at the point $\xi (t)$ a confidence interval is given by the equation $(x-\xi (t))^2 = 2 \, (erf^{-1}(p))^2 \, \delta ^2 \mu (t)$. Hence the boundaries of the $100p \%$-confidence band can be given as $x_{1,2}(t) = \xi (t) \pm erf^{-1}(p) \delta \sqrt{2 \, \mu (t)} u(t)$ where $erf(z) = \frac{2}{\sqrt{\pi }} \int _{0}^{z} e^{-t^2} dt$.