Invariant Distribution in Bu¤er-Stock Saving and Stochastic Growth Models

I derive a simple condition for the existence of a stable invariant distribution of an increasing Markov process on a non-compact state space. I use this condition in two workhorse models in macroeconomics. First, I settle a conjecture of Carroll by characterizing the conditions under which the canonical bu¤er stock saving model has a stable invariant distribution. Second, I show that in the Brock-Mirman one-sector growth model, a stable invariant distribution exists for a large class of production functions, even if the shocks and capital stock can be unbounded. These results help characterize long term behavior in dynamic economies, and have implications for the validity of numerical solutions. E-mail: [email protected]. I thank John Campbell, Chris Carroll, Gary Chamberlain, Drew Fudenberg, Hugo Hopenhayn, Miklos Koren, Alex Michaelides and Ricard Torres for helpful comments, the Social Science Research Council and the Institute for Humane Studies for support. In many stochastic dynamic models in economics, agents’behavior follows a Markov process. In such models, due to unremitting uncertainty the economy never settles down to a deterministic “rest point.”Therefore it is often more convenient to consider a rest point in the stochastic sense, i.e., an invariant distribution. Such a distribution can be interpreted as the long run outcome of the model if it is stable, that is, if the economy eventually converges to it for any initial condition. From this perspective, a stable invariant distribution can be viewed as the “long run equilibrium” in a stochastic dynamic model. This paper studies the existence of a stable invariant distribution for a class of Markov processes de…ned on a non-compact state space. Allowing the state space to be non-compact is important for many economic applications. For example, consider the dynamic macroeconomic models used in consumption theory, economic growth and asset pricing. In many of these models, consumers have power utility preferences and shocks are lognormally distributed, a combination that by construction leads to an unbounded state space. Determining whether a stable invariant distribution exists in these non-compact models is useful both for theoretical and applied reasons. From a theoretical perspective, such results help characterize whether long term behavior is stationary or explosive, which in turn has further economic implications. For an example, consider the bu¤er stock saving model analyzed in more detail below. When this model has a stable invariant distribution, consumption is mean reverting, and hence bu¤er stock behavior obtains (as in Carroll, 1997); in particular, consumption growth is predictable and excessively sensitive to temporary shocks. In contrast, when there is no stable invariant distribution, eventually consumption becomes a random walk (as in Hall, 1978), and therefore, in the long run, the standard permanent income hypothesis holds: consumption is unpredictable and not excessively sensitive. Thus, whether a stable invariant distribution exists has strong implications for long term consumption behavior in this model. From an applied perspective, stable invariant distributions can matter for the validity of numerical solutions. In practice, numerically solving macroeconomic models requires imposing bounds on the realizations of shocks, resulting in a compact state space. For these numerical predictions to approximate the original model, the simulated and the original economies should have similar long term behavior. Typically, compactness implies that a stable invariant distribution exists in the numerically solved economy; hence numerical predictions are likely to be misleading if the original model leads to non-stationary behavior. In this case, the seemingly stationary dynamics in the sim-