Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER VI. THE MATHEMATICS OF HUMAN BEHAVIOR

Does the Consumer Price Index cause sunspots? • Are there optimal marriages? • Game theory or conflict theory? • How much would you pay for a lottery ticket that is expected win a million dollars? • Is it irrational to throw money into the trash bin? • Is someone who believes everything “simple”? • Can one arrive at a decision without preconceptions? • What is evolutionary rationality?

44. MACRO-CONSIDERATIONS

Since the dawn of history man's behavior has been the subject of analysis and debate in various spheres: literature, art, law, and political and philosophical studies. Yet, the use of a mathematical approach to describe and analyze people's conduct and decisions began only toward the end of the eighteenth century. In this chapter we describe some of these developments.

Human conduct, particularly in economic matters, can be divided into individual behavior and group behavior. Clearly the two are connected, as individual behavior determines group behavior. Yet, in economic issues it is still difficult to find a mathematical model that can provide a quantitative prediction of how global economic parameters follow from the decisions of individuals. It was the Scottish philosopher and economist Adam Smith (1723–1790) who coined the phrase the invisible hand. He presented the concept in his book An Inquiry into the Nature and Causes of the Wealth of Nations, published in 1776. In this book Smith laid the foundations of the theory of capitalism: every individual tries to maximize his own welfare, without regard to the needs of the public, and an invisible hand translates those individual actions such that they improve the situation of the society. The nature of the invisible hand remained unexplained. The first explanations did not appear until the 1950s, when economists began systematically to base the theory of capitalism on defined fundamentals; however, this approach met with only very limited success.

On the face of it, behavior in which every individual is concerned only with himself is highly consistent with Darwin's concepts of evolution, as the evolutionary struggle leads to competitive conduct. A closer look, however, reveals that competition in nature is not between individuals but between species. The victorious species are those that survive for generations, and they are not necessarily the species in which every individual fends for itself. A species may be able to survive because its members are prepared to sacrifice themselves for the common good. Such an evolutionary analysis showing the link between individual behavior and the success of the group does not yet exist with regard to economic conduct of communities. Furthermore, the performance of a large economy is largely the result of the decisions of many individual decision makers, each one of whom has little or negligible effect. In this sense there is a similarity with the mathematical description of nature. No quantitative mechanism for the invisible hand has been discovered that combines the elementary particles that have wave characteristics into an element that fulfills Newton's laws.

The mathematical tools used today to analyze macroeconomic activity are not essentially different from those developed for understanding how nature functions. These tools include equations of different types, for example, differential or other equations that deal with economic quantities such as consumption, savings, and interest rates. The equations are meant to describe how an economy works and not what type of economy we might be interested in. An analysis of the model sometimes may help us understand what steps the fiscal or monetary policy makers ought to take to achieve the desired objective. The model itself, however, describes the economy as it is. As in a social world, we cannot perform proper controlled experiments, so economists use data provided by bureaus of statistics. The technique used to analyze these parameters is known as econometrics, which is a development of the statistics that we described in the previous chapter. The methods developed for purposes of economic analysis are very advanced, but the approach is not essentially different than that which developed over the years for use in the natural sciences and technology. Success in describing macroeconomic conduct is lagging behind the level of the success of mathematics in describing physics and its uses in technology. Is this merely a question of time, and will the gap be closed with the improvement of the existing models, or is there possibly a need for a new mathematics to describe human behavior? There is no unequivocal answer.

We will not describe the macroeconomic models in detail. We will just present two examples of considerations specific to social sciences. Both are related to the recipients of the Nobel Prize in Economics in 2011. The official name of the prize is the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (the Sveriges Riksbank is the central bank of Sweden). Alfred Nobel did not specify the social sciences as one of the fields in which a prize would be awarded. We cite these examples because of the special way in which mathematics is used to describe the complexity of human conduct, and they do not reflect the entire range of uses of mathematics in macroeconomics.

An inherent element of human decision making is the assessment of what is likely to happen in the future, when in many cases the individual considers that his influence on the future is negligible. For years there was an understanding that macroeconomic developments are affected by individuals’ expectation regarding the future, but that understanding was not translated into a component in equations. Robert Lucas of the University of Chicago, who was awarded the Nobel Prize in Economics in 1995, and colleagues who included Thomas Sargent of New York University, winner of the Nobel Prize in Economics in 2011, developed the theory of rational expectations. They found a mathematical way to incorporate market expectations in an equation determining the process of development of the economic parameters. These expectations are part of the variables in the model; they influence the other variables and are influenced by them. Market expectations that affect the development of the market are clearly specific to social sciences. The formulation that describes in mathematical terms the role of such expectations in future developments was adopted by economists and constitutes an integral element in many macroeconomic models.

The second example is merely anecdotal and should not be considered as representing econometric practice. We chose this example because it teaches us something about the link between mathematics and uses. The joint winner of the Nobel Prize in Economics in 2011 with Thomas Sargent was Christopher Sims of Princeton University. The prize citation refers to Sims's contribution to the analysis of time series, that is, statistical series that change with time. The analysis of statistical series in general, and those that change with time in particular, have long interested scientists, and the basis for the mathematical methods of analyzing the errors in these series dates back as early as in the time of Carl Friedrich Gauss. Among other advances, systems were developed for finding whether two series of data were correlated, and that gave quantitative indices for the degree of correlation. In their use in the natural sciences, however, the question did not arise as to which of the two series was dominant, that is, which caused changes in the other series. For example, there is a causal relationship between the Earth's revolving on its own axis and high tide and low tide. Yet no one looked in the two data series for the answer to the question of what is the cause and what is the effect, does the tidal flow cause the Earth to revolve or vice versa. The answer is derived from the laws of nature themselves. In general, in natural sciences we do not try to derive the cause and effect from the two data series themselves, but we try to derive them from their underlying model. Unfortunately, the mathematical models of social and economic occurrences are not reliable enough to enable a similar analysis to be performed. It is natural, therefore, that attempts will be made to derive the cause and determine which of the two series is dominant from the series themselves. Among Sims's contribution to understanding time series was his enhancement of a method proposed by the British economist Clive Granger (1934–2009), a 2003 Nobel laureate in economics. The method was supposed to determine which of two data series was the cause of the other. The test is known as the Granger-Sims causality test and serves to test and examine causality in many areas of social science and economics.

In 1982 two economists, Richard Sheehan and Robin Grieves, published results of the use of the Granger-Sims causality test to examine possible causality between the appearance of sunspots and the business cycle in the US economy, related to both gross national product (GNP) and the price index. The article was published in the Southern Economic Journal (volume 48, pages 775–78). The results were statistically significant and showed that the business cycles in the American economy is the cause of the sunspots. It is clear that this result is inconceivable. Yet to make it quite clear, it does not disqualify the statistical test. What should be learned from it is that we should not rely on a statistical test that is not supported by an independent model. As there is no model that incorporates the effect of GNP on sunspots, the statistical analysis that examines this is not applicable. Such use of a statistical test is limited. The right approach is first to propose a model representing a causal relation, and then to leave it to statistics to confirm or refute the model. A statistical test alone without a possible model of the effect itself can lead to fundamental errors.