On the Nonlinear Dynamics of Models with a Cash-in-advance Constraint

Recently there has been a growing interest in dynamic general equilibrium models exhibiting indeterminacy and nonlinear endogenous fluctuations. In these models, "complex" dynamics typically arise as a consequence of increasing returns-to-scale, often triggered by positive external effects (see e.g. Benhabib and Farmer [1994]). While early results relied on empirically unrealistic scale economies, more recent researches have demonstrated that the magnitude of increasing returns giving rise to complex dynamics (local indeterminacy and bifurcations) is consistent with recent empirical estimates. Even though these results improve the plausibility of endogenous fluctuations, it should be noticed that they depend to a great extend on the shape of the utility function : (1) the intertemporal elasticity of substitution in consumption (IES) must be significantly greater than 1. Yet, this requisite is at odds with the empirical evidence which suggests that the IES is much lower than unity, many estimates beeing indeed below 0.5 (see e.g. Hall [1988]). (2) Most of these results have been established under the assumption that the labor supply is infinitely elastic (indivisibility of labor assumption). Yet, the few robustness exercises performed indicate that slightly lowering the labor supply elasticity strikingly increases the required degree of increasing returns. In this paper I examine the dynamical properties of MONETARY (cash-in-advance) general equilibrium models with elastic labor supply. As, in particular, I ask whether increasing returns are needed for endogenous fluctuations to emerge, I dismiss external effects and consider a constant returns-to-scale technology. Due to the cash-in-advance constraint the equilibrium dynamical system is THREE dimensional (except if the static preferences are separable and logarithmic in consumption) "Graphical" arguments are then used to study the properties of such a 3D system. I show that : (1) if, as commonly assumed, the preferences are separable, the model exhibits indeterminacy, hence sunspots equilibria, regardless of the elasticity of labor supply and for values of the IES in accordance with the bulk of empirical estimates. (2) if the preferences are NON separable and if the marginal utility of consumption is a decreasing function of the labor effort, the model exhibits indeterminacy, 2-period cycles (Flip bifurcation) and quasi- periodic fluctuations (Hopf bifurcations).