Study on Bubbles Solutions of the Cagan Model

The purpose of this paper is to study bubbles solutions to the Cagan hyperinflation models under rational expectations. Both the price and exchange rate bubbles are considered. Specifications of the Cagan model under rational expectations will be briefly described, in which the price and exchange rate series are expressed in first-order linear difference equations. The particular and the homogenous solutions to the Cagan model can then be derived. The particular or fundamental solution characterizes a unique dynamic movement of an underlying fundamental process. Several representations of the fundamental solution will be explored. The homogenous or bubble solution is non-unique in a rational expectations framework. Some examples of bubble solution with different dynamic properties are specified. Also, examples of bursting bubble specifications will be illustrated. It is concluded that the problems of multiple solutions make indirect tests more attractive than direct tests for bubble detection. In addition, the general solution, which is just the sum of particular and homogenous solutions, will be discussed. Hence, the bubble paths are characterized as any deviations of the general solution from the fundamental solution when the model is specified correctly.