Survey_AEH_2003

Abstract

In the standard hyperinflation model real money demand is a linearly decreasing function of expected inflation where md t denotes real money demand and xt+1 the inflation factor from t to t + 1. Here E? t xt+1 denotes expected inflation, which we do not restrict to be fully rational (we will reserve Etxt+1 for rational expectations). Money demand functions of the above form can be generated by monetary overlapping generations models with log-utility functions. Alternatively they can be viewed as a log-version of the (Cagan 1956) demand function. We first consider the situation where agents have information about all variables up to time t and wish to learn the parameters of the rational expectations solution (4). As is well-known, the conditions for local stability under adaptive learning are given by expectational stability (E-stability) conditions. Therefore, we first discuss the E-stability conditions for the REE, after which we take up real time learning. Next we consider real time learning of the set of ARMA equilibria (4). This section shows that stochastic approximation theory can be applied to show convergence of least squares learning when the PLM of the agents has AR(1) form and the economy can converge to the locally determinate AR(1) equilibrium (6). For technical reasons the stochastic approximation tools cannot be applied for the continuum of ARMA(1,1)-REE. Therefore, real time learning of the class (4) REE will be considered in section 7 using simulations. The observability of current states, as assumed in the previous section, introduces a simultaneity between expectations and current outcomes. Technically this is reflected in xt appearing on both sides of the equation when substituting the PLM (7) into the model (3). To obtain the ALM one first has to solve this equation for xt. Although this is straightforward mathematically, it is not clear what economic mechanism would ensure consistency between xt and the expectations based on xt. Moreover, in the non-linear formulation there may even exist multiple mutually consistent price and price expectations pairs, as pointed out in (Adam 2003). To study the role of the precise information assumption, we introduce a fraction of agents who cannot observe the current state xt. effect must learn to make forecasts that are consistent with current outcomes, which allows us to consider the robustness of the preceding results.