Sunspots and Non-Linear Dynamics

The difficulties associated with the appraisal of the determinacy properties of a three-dimensional system are circumvented by the introduction of a new geometrical argument. It brings about a complete typology of the eigenvalues moduli in discrete time three-dimensional dynamical systems and then provides a new apparatus for assessing from a geometrical standpoint the emergence of local bifurcations for parameterised economies. The argument is considered through the extensive characterisation of the stability properties of a benchmark model of inter-temporal economic analysis.

We present conditions for the emergence of singularities in DGE models. We distinguish between slow-fast and impasse singularity types, review geometrical methods to deal with both types of singularity and apply them to DGE dynamics. We find that impasse singularities can generate new types of DGE dynamics, in particular temporary determinacy/indeterminacy. We illustrate the different nature of the two types of singularities and apply our results to two simple models: the Benhabib and Farmer (1994) model and one with a cyclical fiscal policy rule.

We analyze a version of the Benhabib and Farmer (1996) two-sector model with sector-specific externalities in which we consider a class of utility functions inspired from the one considered in Jaimovich and Rebelo (2009) which is flexible enough to encompass varying degrees of income effect. First, we show that local indeterminacy and sunspot fluctuations occur in 2-sector models under plausible configurations regarding all structural parameters—in particular regarding the intensity of income effects. Second, we prove that there even exist some configurations for which local indeterminacy arises under any degree of income effect. More precisely, for any given size of income effect, we show that there is a non-empty range of values for the Frisch elasticity of labor and the elasticity of intertemporal substitution in consumption such that indeterminacy occurs. This contrasts with the results obtained in one-sector models in both Nishimura et al. (2009), in which it is shown that indeterminacy cannot occur under either GHH and KPR preferences, and in Jaimovich (2008) in which local indeterminacy only arises for intermediary income effects.

This essay links some of my own work on expectations, learning and bounded rationality to the inspiring ideas of Jean-Michel Grandmont. In particular, my work on consistent expectations and behavioral learning equilibria may be seen as formalizations of JMG’s ideas of self-fulfilling mistakes. Some of our learning-to-forecast laboratory experiments with human subjects have also been strongly influenced by JMG’s ideas. Key features of self-fulfilling mistakes are multiple equilibria, excess volatility and persistence amplification.

This paper shows that regime-switching sunspot equilibria easily arise in a one-sector growth model with aggregate decreasing returns and arbitrarily small externalities. We construct a regime-switching sunspot equilibrium under the assumption that the utility function of consumption is linear. We also construct a stochastic optimal growth model whose optimal process turns out to be a regime-switching sunspot equilibrium of the original economy under the assumption that there is no capital externality. We illustrate our results with numerical examples.

We study a two-sector \(\textit{OLG}\) economy in which a share of old age consumption expenditures must be paid out of money balances and we appraise its dynamic features. We first show that competitive equilibrium is dynamically efficient if and only if the share of capital on total income is large enough while a steady state capital per capita above its Golden Rule level is not consistent with a binding liquidity constraint. We thus focus on the gross substitutability in consumption and on dynamic efficiency assumptions and show that, gathered together, they ensure the local determinacy of equilibrium and, as a consequence, rule out sunspot fluctuations. In addition, we prove that the unique steady state may change its stability from a saddle configuration to a source one (undergoing a flip bifurcation) for a capital intensive investment good as well as for a capital intensive consumption good, when the elasticity of the interest rate is set low enough. However, when the investment good is not too capital intensive, the flip bifurcation turns out to be compatible with high elasticities of the interest rate too. Analogous results within dynamic efficiency are found in the non-monetary model, the existence of a flip bifurcation requiring now a capital intensive investment good. Eventually, under dynamic inefficiency, in the non-monetary economy local indeterminacy may instead appear, either through a Hopf bifurcation or through a flip one, and its scope improves as soon as the consumption good becomes more and more capital intensive.

We treat a three-dimensional continuous-time model that includes one predetermined variable and two non-predetermined variables. We assume (1) that the model has a two-dimensional well-located invariant manifold and (2) that the manifold includes a one-dimensional closed curve that could be either a homoclinic orbit or a closed orbit. We construct a stationary sunspot equilibrium in this three-dimensional model by means of generalizing the methods due to Nishimura and Shigoka (2006) and Benhabib et al. (2008). By appealing to the same argument as in Nishimura and Shigoka (2006) we can apply our result to some variants of the Lucas (1988) model the transitional dynamics of which is three-dimensional and undergoes homoclinic bifurcation.

This paper considers rational land and housing bubbles in an infinite-horizon general equilibrium model. Their demands rest on two different grounds: the land is an input to produce while the house may be consumed. Our work differs from the existing literature in two respects. First, dividends on both these long-lived assets are endogenous and their sequences are computed. Second, we introduce and study different concepts of bubbles, including individual and strong bubbles.

We explore the stabilizing role of monetary policy on the existence of endogenous fluctuations when the economy experiences a rational bubble. Considering an overlapping generations model, expectation-driven fluctuations are explained by a portfolio choice between three assets (capital, bonds and money), credit market imperfections and a collateral effect. They occur under a positive bubble on bonds. The key mechanism relies on the existence of gaps between the returns on assets due to financial distortions. Then, we study the stabilizing role of the monetary policy. Such a policy managed by a (standard) Taylor rule has no clear stabilizing virtues.

We discuss the stabilization role of consumption taxes under a balanced-budget rule in the presence of consumption externalities of the “keeping up with the Joneses” type. We consider a finance constrained economy and depart from a situation where sufficiently strong externalities make the steady state indeterminate, if government intervention is absent. Sufficiently procyclical consumption tax rates are able to ensure local saddle path stability. However, this procyclicality leads to the appearance of another steady state with lower levels of output which is a source or indeterminate. Therefore, government intervention with stabilization purposes may not be successful.

We construct a simple neoclassical model to capture the Keynesian idea that equilibrium aggregate supply is determined by aggregate demand and thus influenced by consumer sentiments about aggregate income. This result is nontrivial because in a standard neoclassical setting it is the aggregate supply (productive capacity) that determines aggregate demand, instead of the other way around, through factor-income shares and market-clearing mechanisms. However, we show that when firms’ production and employment decisions must be based on expectations of aggregate demand and that realized demand follows from firms’ production and employment decisions through market-clearing mechanisms, rational expectations about aggregate demand can lead to stochastic sentiment-driven equilibria despite the absence of production externalities, incomplete financial markets, strategic complementarity or any non-convexities in the model. The key is imperfect information arising naturally from the fact that production and employment decisions by firms, and consumption and labor supply decisions by households, are all made prior to goods being produced and exchanged, and before market clearing prices are realized. The sentiment-driven equilibria in our model are not based on randomizations over multiple fundamental equilibria as in many sunspot-driven business cycle models.

For a neoclassical two-sector optimal growth model with Cobb-Douglas technologies and sector specific technical progress, we examine three properties: (i) each sector has an optimal path by which it will grow at a constant growth rate (an optimal constant growth path); (ii) the optimal constant growth paths satisfy saddle path stability; (iii) the elasticity of substitution between total labor and total capital is less than one along the optimal constant growth paths. These results, presented by Acemoglu and Guerrieri in their model with two intermediate good sectors and one final good sector, will give a firm theoretical base for establishing the Kaldor and Kuznets facts, and are proven herein for a neoclassical growth model.

In a dynamic economy, money provides liquidity as a medium of exchange. A central bank that sets the nominal rate of interest and distributes its profit to shareholders as dividends is traded in the asset market. A nominal rates of interest that tend to zero, but do not vanish, eliminate equilibrium allocations that do not converge to a Pareto optimal allocation.

We analyze an economy with taxes and transfers denominated in dollars and an information friction. It is the information friction that allows for volatility in equilibrium prices and allocations. When the price level is expected to be stable, the competitive equilibrium allocation is Pareto optimal. When the price level is volatile, it is not Pareto optimal, but the stable equilibrium allocations do not necessarily dominate the volatile ones. There can be winners and losers from volatility. We identify winners and losers and describe the effect on them of increases in volatility. Our analysis is an application of the weak axiom of revealed preference in the tax-adjusted Edgeworth box.

We recast the well known no trade result in Milgrom and Stokey (1982) using the appropriate definition of efficiency among several available in the asymmetric information framework.