# Non-Self-Averaging Phenomena in Macroeconomics: Neglected Sources of Uncertainty and Policy Ineffectiveness

Masanao Aoki

## Introduction

The main analytical exercise by mainstream macroeconomists such as Romer (1986), Lucas (1988), Grossman and Helpman (1991), and Aghion and Howitt (1992) is to explicitly consider the optimization by representative agents using quadratic criterion functions in such activities as education, on-the-job training, basic scientific research, and process and product innovations. This approach is found not only in the study of economic growth but also in research on business cycles. In this chapter we argue that this research program, which dominates modern macroeconomics, is misguided because it does not recognize the distinction between self-averaging and non-self-averaging aspects of their optimization problems. Whether in growth or business cycle models, the fundamental motivation for often complex optimization exercises is that they are expected to lead us to a better understanding of dynamics of the *mean *or *aggregate* variables. The standard procedure is to begin with an analysis of optimization for the representative agent and then to translate it into an analysis of the economy as a whole. These exercises presume that different microeconomic shocks and differences among agents will cancel out in the means, and the results can be well captured by the analysis of the representative agents.

We show that the phenomenon of non-self-averaging has material consequences for macroeconomic policy development. Specifically, models that exhibit non-self-averaging—that is, those whose standard deviations divided by the means do not decrease as the systems grow—are ubiquitous, and macroeconomic simulations using them can give rise to uninformative or misleading policy results.

More formally, given a macroeconomic model, some random variable *X _{n}* of the model, where

*n*is model size, is called self-averaging if its coefficient of variation (CV), that is, the ratio of standard deviation of

*X*divided by its mean, tends to zero as

_{n }*n*goes to infinity, and non-selfaveraging if the limit is non-zero or if the limit tends to infinity.

This phenomenon is related directly to the magnitude of economic fluctuations and is consistent with the size and scaling of fluctuations observed both recently and in the more distant past.

By way of examples, we show how macroeconomic policy can be rendered totally ineffective solely as a result of non-self-averaging. Altogether three types of non-self-averaging models are discussed: two-parameter Poisson-Dirichlet models; urn models; and two blocks of interdependent macroeconomic models (see Aoki, 2008a; 2008b; 2008c). After a brief introduction to non-self-averaging, we present analytical results on policy ineffectiveness by means of two simple introductory examples. These show (a) the importance of coefficients of variation, (b) how coefficients of variation enter into a well-known economics problem, and (c) that if the coefficient of variation becomes large then policy becomes ineffective. A connection with the two- parameter Poisson-Dirichlet model is then established to show how GDP becomes non-self-averaging. The chapter then discusses relations of urn models to macroeconomic models and exhibits examples of urn models in the literature that are non-self-averaging.