Aggregation, Persistence and Volatility in a Macro Model

Starting from microeconomic foundations, we derive a general formula for the aggregation of outputs of heterogeneous firms (or sectors), and we solve explicitly for the fundamental intertemporal equilibrium path of the aggregate economy. The firms are subject to temporary technology shocks, but the aggregate output has radically different dynamical properties, and a special form of long memory and nonlinearity never used hitherto. We study, analytically, the implied time series properties of the new process characterizing aggregate GDP per capita. This process is more persistent than any dynamically-stable linear process (e.g. autoregressions) and yet is mean-reverting (unlike unit-root processes), and its volatility is of a greater order of magnitude than that of any of its components. This amplification of volatility means that even small shocks at the micro level can lead to large fluctuations at the macro level. The process is also characterized by long cycles which have random lengths and which are asymmetric. Increased monopoly power will tend to reduce the amplitude and increase the persistence of business cycles. Strikingly, we find that the nonlinear aggregate process has an S-shaped decay of memory, similar to the data but unlike linear time series models such as the widely-used Auto-Regressive Integrated Moving-Average (ARIMA) processes and their special cases (including fractional Integration).