Rational and Near-Rational Bubbles Without Drift

Abstract

This paper derives a general class of intrinsic rational bubble solutions in a standard Lucas-type asset pricing model. I show that the rational bubble component of the price-dividend ratio can evolve as a geometric random walk without drift, such that the mean of the bubble growth rate is zero. Driftless rational bubbles are part of a continuum of equilibrium solutions that involve an explicit trade-off between the mean and volatility of the bubble growth rate. I also propose a near-rational asset pricing solution in which the representative agent does not construct separate forecasts for the fundamental and bubble components of the asset price. Rather, the agent constructs only a single forecast for the total asset price that is similar in form to the corresponding rational forecast, but involves fewer parameters. The parameters of the agent's forecast rule are chosen to match the moments of observable data. In the near-rational equilibrium, the actual law of motion for the price-dividend ratio is stationary, highly persistent, and nonlinear. The agent's forecast errors exhibit near-zero autocorrelation at all lags, making it difficult for the agent to detect a misspecification of the forecast rule. Unlike a rational bubble, the near-rational solution allows the asset price to occasionally dip below its fundamental value. Under mild risk aversion, the near-rational solution generates pronounced low-frequency swings in the price-dividend ratio, positive skewness, excess kurtosis, and time-varying volatility - all of which are present in long-run U.S. stock market data. An additional contribution of the paper is to demonstrate an approximate analytical solution for the fundamental asset price that employs a nonlinear change of variables.