Fortunately, research in semantic information and semiotics is advancing. For example, machine translation allows a computer to select the meaning of a word by looking at context, at adjacent words. The user asks, “What is the longest river in South Africa?” and Siri provides a direct verbal answer (“The longest river in South Africa is the Orange River”). Such search is now becoming well established around the world.
However, semantic search may take a long time to reach the human mind’s abilities to understand narratives. In the meantime, researchers can still quantify the study of narratives by using multiple research assistants who receive explicit instructions to read narratives and to classify and quantify them according to their essential emotional driving force. Advances in psychology, neuroscience, and artificial intelligence will also improve our sense of structure in narrative economics. Companies like alexability. com (Alexandria), alpha-sense. com, prattle.co, and quid. com are beginning to offer intelligent searches of public documents and the media that could help organize information about shared narratives.
As research methods advance, and as more social media data accumulate, textual analysis will become a stronger force in economics. It may allow us to move beyond 1930s-style models of income-consumption feedback and Keynesian multipliers that are still influential today and get closer to all the kinds of feedback that drive economic events. It will also help us better understand the deliberate manipulations and deceptions we have experienced, and it will help us formulate economic policies that take narratives into account.
We should be looking forward to better understanding the patterns of human thinking about the forces that cause economies to boom at times and to stagnate at others, to go through creative times and backward times, to go through phases of compassion and phases of conspicuous consumption and self-promotion, to experience periods of rapid progress and periods of regression. I hope this book confirms the possibility of real progress in getting closer to the human reality behind major economic events without sacrificing our commitment to sound scholarship and systematic analysis.
Appendix: Applying Epidemic Models to Economic Narratives
Epidemiology, a subfield of medicine, developed most productively in the twentieth century. Its greatest contribution, a mathematical theory of disease epidemics, sheds powerful light on idea epidemics as they influence economic events. We can adapt this theory to model the spread of economic narratives.
A Theory of How Disease Spreads
The mathematical theory of disease epidemics was first proposed in 1927 by William Ogilvy Kermack, a Scottish biochemist, and Anderson Gray McKendrick, a Scottish physician. It marked a revolution in medical thinking by providing a realistic framework for understanding the dynamics of infectious diseases.
Their simplest model divided the population into three compartments: susceptible, infective, and recovered. It is therefore called an SIR model or compartmental model. S is the percentage of the population who are susceptible, people who have not had the disease and are vulnerable to getting it. I is the percentage of the population who have caught the disease and are infective, who are actively spreading it. R is the percentage of the population who are recovered, who have had the disease and gotten over it, who have acquired immunity, and who are no longer capable of catching the disease again or spreading it. Nobody dies in this original model. The sum of the percentages is 100%, 100% = S + I + R, and the population is assumed constant.
According to the Kermack-McKendrick mathematical theory of disease epidemics, in a thoroughly mixing constant population the rate of increase of infectives in a disease epidemic is equal to a constant contagion parameter c times the product of the fraction of the total population who are susceptible S and the fraction infective I, minus a constant recovery rate r times the fraction of infectives I. Each time a susceptible person meets an infective person, there is a chance of infection. In a large population, the chance averages out to a certainty. The number of such meetings per unit of time depends on the number of susceptible-infective pairs in the population, hence the product SI.1 The three-equation Kermack-McKendrick SIR model is:
There is no algebraic solution to this model, only approximations.2 Similar equations also appear in chemistry, where they are called rate equations or consecutive chemical reactions.3
In the model used in this book, the contagion rate is cS, the product of a constant contagion parameter c and the time-varying fraction of susceptible people S. The recovery rate is constant, r. If we divide both sides of the second equation by the fraction of infective people I, we can see that the second equation is nothing more than a statement that the growth rate of the fraction of the population who are infectives is equal to the contagion rate cS minus the recovery (or forgetting) rate r. This conclusion makes sense: if it is to grow, the epidemic has to be spreading faster than people are recovering, and it is common sense that the contagion rate should depend on the fraction of the population susceptible to infection.
The first and third equations are very simple. The first equation says that the number of susceptibles falls by one with every new infection, because a susceptible turns into an infective. The third equation says that the number of recovereds rises by one with every new recovery, because when a person recovers from the illness (or in our context forgets a narrative) an infective turns into a recovered. We will see below that this elementary model, which carries an essential insight about the path of epidemics, can be modified to include a growing population and many other factors specific to a particular epidemic.
FIGURE A.1. Theoretical Epidemic Paths
Solution to Kermack-McKendrick SIR model for I0 = .0001%, c = .5, r = .05. The heavy bars show the percentage of the population who are sick and spreading the disease. The model assumes no medical intervention; the epidemic ends on its own, even though there are still susceptibles in the population, and not everyone was ever infected. Source: Author’s calculations.
Figure A.1 shows an example, implied by the above three equations, where one person in a million is initially exposed, I0 = 0.0001%, and parameters c = .5, r = .05. In this case, almost 100% of the population eventually gets infected. During a disease epidemic, the public tends to focus on the infectives, the bell-shaped curve in the figure. Attention also focuses on the number of newly reported cases, the speed of transition from susceptible to infective, which follows a similarly shaped bell-shaped curve if r is not too far below c. For narratives, we compare plots of counts of words and publications to the infective curve as in the figure.
The SIR model implies that from a small number of initial infectives, the number of infectives follows much the same hump pattern, from epidemic to epidemic, rising at first, then falling. A mutation in an old, much-reduced disease may produce a single individual who is infective with the new strain. Then there will be a lag, possibly a long lag if c is small, before the disease has infected enough people to be noticed in public. The epidemic will then rise to a peak. Before everyone is infected, the epidemic will then fall and come to an end without any change in the infection or recovery parameters c and r.
Not everyone will catch the disease. Some people escape the disease completely because they do not have an effective encounter with an infective. The environment gradually becomes safer and safer for them because the number of infectives decreases as they get over the disease and become immune to it. Thus there are not enough new encounters to generate sufficient new infectives to keep the disease on the growth path. Eventually, the infectives almost disappear, and the population consists almost entirely of susceptible and recovered. Applying this model to narratives: because not everyone is infected, some people will say after an economic narrative epidemic that they never even heard of the narrative, and they will be skeptical of its influence on the economy even if the narrative is indeed very important to economic activity.
Which factors combine to spread a major disease that ultimately reaches a lot of people (the total fractio
n of the population ever infected and recovered)? The disease’s reach is determined by the ratio c/r. As time goes to infinity, the fraction of people who have ever had the disease goes to a limit R∞ (called the size of the epidemic) strictly less than 1. It follows directly from the first and third equations that Given the initial condition on the fraction of the population initially infected I0 that , and because I∞ = 0, 1 = S∞ + R∞, we have:
which provides the relationship between the ultimate number ever infected by the disease and c/r. If we could choose c and r, we could make the size of the epidemic R∞ anything we want between I0 and 100%. If we define “going viral” as , then we see a viral event happening from I0 close to zero when . If we multiply both parameters, c and r, by any positive constant a, then the same three equations are satisfied by S(at), I(at), R(at).
Higher c/r corresponds to higher size of epidemic R∞, regardless of the level of c or r, while higher c itself, holding c/r constant, yields a faster epidemic. For an epidemic to get started from very small beginnings, when S is close to 1, c/r must be greater than 1. Depending on the two parameters c and r, there can be both fast and slow epidemics that look identical if the plot is rescaled. If we also vary the ratio c/r, we can have epidemics that play out over days and reach 95% of the population, or epidemics that play out over decades and reach 95% of the population, or epidemics that play out over days and reach only 5% of the population, or epidemics that play out over decades and reach 5% of the population. But in each case, we can have hump-shaped patterns of infected that on rescaling look something like the heavy line in Figure A.1.
Variations on the SIR Model
The Kermack-McKendrick SIR model is the starting point for mathematical models of epidemics that have, over the better part of a century since, produced a huge literature. Among the different versions, the basic compartmental model has been modified to allow for gradual loss of immunity, so that recovereds are gradually transformed into susceptibles again (the SIRS model).4 The SIR model can also be modified so that an encounter between a susceptible and an infected leads to an increase in exposed E, a fourth compartment who become infected later (the SEIR model). The model has also been modified to incorporate partial immunity after cure, birth of new susceptibles, the presence of superspreaders with very high contagiousness, and geographical patterns of spread.
These models, with modifications appropriate for the disease studied, have been useful for predicting the course of epidemics. For example, the SEIR model has been modified to explain the spread of influenza geographically with the assumption that the exposed but still asymptomatic are capable of long-distance travel. Applying the model to influenza data and data on intercity volume of air transportation, R. F. Grais and her coauthors found that their model helps explain intracity and intercity time patterns of influenza outbreak.5
Another compartmental model example is a stochastic extension of an SEIHFR model, where S is susceptible, E is exposed, I is infected, H is hospitalized, F is dead but not buried, and R is recovered or buried. This model has been fitted to data on African Ebola epidemics,6 and it takes into account public efforts to stem contagion of the disease through hospitalization and proper disposal of bodies.
The SEIHFR compartmental model has six compartments, but future models of economic narratives might well benefit from even more compartments. For example, a model for the spread of the technological unemployment narrative (see chapter 13) might include separate compartments for unemployed and infected and unemployed and uninfected, employed and infected and employed and uninfected, as well as extra equations that come from conventional economic models.
Economic models might also take inspiration from the medical literature on co-epidemics to incorporate contagious economic narratives into economic models. In a medical setting, a co-epidemic occurs when the progress of one disease interacts with the progress of another. For example, HIV and tuberculosis have been identified as coinfective: many more people have both diseases than would be predicted by two independent epidemic models. Elisa F. Long and her coauthors (2008) have proposed a variation of the basic compartmental model along Kermack-McKendrick lines that allows for people infected by one of these diseases to be more likely to catch and spread the other.7 Models like this one could represent narrative constellations in which multiple narratives support one another by contagion. Such models could also represent the interaction of economic narratives, such as the technological unemployment narrative, with economic status, such as unemployment.
Structural macroeconomic models commonly include simple univariate autoregressive integrated moving average (ARIMA) models to represent error terms or driving variables for which there is no economic theory. George E. P. Box and Gwilym Jenkins first popularized the ARIMA models in a 1970 book. While Box and Jenkins described these models as useful in any realm of science, economists have used them most aggressively.8 Owing to a well-developed theory of forecasting of times series that can be described in ARIMA terms, the epidemic among economists of ARIMA models led to a slightly delayed epidemic of rational expectations models, which peaked (according to Google Ngrams) around 1990 but still remains prevalent today. The ARIMA models are an alternative to the compartmental models described in this appendix. But there is something essentially arbitrary about the ARIMA models, which, unlike the compartmental epidemic models, lack a theoretical underpinning.9
The ARIMA methods can be improved with the theoretical epidemic models, using a combination of simulation, classification, statistical and optimization techniques to forecast the epidemic curve when contagion rates and recovery rates vary through time.10 We can selectively bring in data other than data on the epidemic itself based on our knowledge of the structure of epidemics, and this takes us well beyond the mindless search for “leading indicators.”
Not all data on epidemics fit the compartmental model framework well. Consider the long-slow US epidemic of poliomyelitis enterovirus cases from the late nineteenth century to their peak in 1952, superimposed on seemingly random one-summer epidemics. A gradual trend toward better cleanliness and hygiene should have had the effect of reducing the incidence of the disease, not increasing it. Paradoxically, the lower incidence of the disease, which was in most cases benign, had the effect of making reported cases involving paralysis or other consequences more common because nursing infants were less likely to receive antibodies from their mothers, which would have helped them gain immunity to the disease’s severe consequences in later reinfections.11
When we apply the compartmental model to social epidemics and to epidemics of ideas, certain changes seem natural. One thought is that the contagion rate should decline with time, as the idea becomes gradually less exciting. One way of modeling that notion comes from Daryl J. Daley and David G. Kendall (1964, 1965), who said that the Kermack-McKendrick model could be altered to represent the idea that infectives might tend to become uninfective after they meet another infective person or a recovered person, because they then think that many people now know the story. Because the story is no longer new and exciting, the newly uninfected choose not to spread the epidemic further.
D. J. Bartholomew (1982) argued that when we apply variations of the Kermack-McKendrick model to the spread of ideas, we should not assume that ceasing to infect others and forgetting are the same thing. Human behavior might be influenced by an old idea not talked about much but still remembered, or “behavioral residue” (Berger, 2013).
There is now a substantial economics literature on network models, including the recent The Oxford Handbook of the Economics of Networks (Bramoullé et al., 2016). There are only a few behavioral epidemic models. The word narrative does not appear even once in the Handbook. Some of these modified SIR models involve complex patterns of outcomes and sometimes cycles. Geographic models of spread are increasingly complicated by worldwide social media connections.12
Some SIR models dispense with the idea of random mixing and choose instead a network structure.13
There may be strategic decisions whether to allow oneself to be infected, and the fraction of the population infected may enter into the decision (Jackson and Yariv, 2005). Other models describe individuals as adopting a practice not merely through random infection but rather through rational calculations of the information transmitted through their encounters with others.14
The core Kermack-McKendrick model may apply no matter how people connect with one another despite concerns that modern media (especially the Internet) make the original SIR model less accurate in describing social epidemics. For this reason, using the SIR model to explain the spread of ideas or narratives may require modifying it to take account of contagion by broadcast as well as contagion through person-to-person contact.15 The existing model can accommodate that change with higher contagion rates for narratives owing to social media automatically directing narratives to people with likely interest in them, regardless of their geography.
Sociologists Elihu Katz and Paul F. Lazarsfeld in 1955 showed impressive evidence for a “two-step flow hypothesis” that cultural change begins with the news media but is completed via the “relay function” of word of mouth within primary groups, led by the relatively few group members who pay attention to the news.16 The marketing profession has responded by promoting word-of-mouth seeding strategies and television ads that feature actors portraying people with whom the common person can identify and simulating direct interpersonal word of mouth. Moreover, marketing literature finds that direct word-of-mouth communications still beat other forms of communication in terms of persuasiveness.17 In considering whether the Internet and social media affect the SIR model, Laijun Zhao and coauthors (2013) argue for a modified SIR model where the news media increase analogues to the parameters c and r.
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