29. Opioids, COVID-19, and Inequality
Everything is complicated; if that were not so, life and poetry and
everything else would be a bore.
—Wallace Stevens
In this final chapter, we apply many-model thinking to three salient policy
issues: the opioid epidemic, the COVID pandemic, and economic
inequality. We show how by engaging multiple models, we can better
reason through these issues and better communicate the challenges that
each presented. We can also see how, particularly in the case of COVID,
that had the public and policymakers engaged multiple models, the costs of
the pandemic would have been less severe. That said, we should not
oversell the potential for models to avoid disaster.
Our treatment of the three cases differs. When analyzing the opioid
epidemic, we include only the barest of details. We do not gather data or
calibrate any models. Rather, we apply the models qualitatively to gain
insights. This more superficial analysis serves as a template for how to
apply many models when reasoning through policies or actions. On the
COVID pandemic, we take a much deeper dive into the models applied and
developed by researchers around the world. That analysis highlights the
seven uses of models introduced early in the book. Finally, our analysis of
income inequality, on the other hand, connects more tightly to the academic
literature than to current policy concerns. Whereas many of the COVID
models were built on similar foundations, our foray into inequality reveals
a variety of models with unique sets of assumptions and different types of
actors.
In all three cases, thinking with many models adds to our knowledge base
and makes us wiser. That wisdom has limits. Given the complexity of these
systems, even if guided by many models, as individuals and collectives, we
will not reach full understandings, always take the appropriate actions, or
design optimal policies. We will make mistakes. All models are wrong. The
map is not the territory. We can learn from our models’ shortcomings to
build more and better models and become even wiser. The path toward

wisdom is paved with humility and a willingness to see our world through
new and different lenses.

Many Models and the Opioid Epidemic
We begin with a many–model analysis of the opioid epidemic. To give
some sense of its scale, according to one estimate in 2015, 4% of the
population of the state of Massachusetts above age eleven had an opioid
use disorder. Nationwide, in 2016, doctors wrote more than 200 million
prescriptions for opioids, between 10 and 12 million people misused
opioids, over 2 million people were classified as having an opioid use
disorder, and more than 30,000 people died from opioid-related causes.
The primary reason that so many opioids were prescribed was that they
work: they reduce pain. Given the 100 million Americans with chronic
pain, opioids had an enormous potential market. The danger with opioids
was, of course, the potential for people to become addicted. To make sense
of how opioids received approval and how the epidemic arose, we apply
four models to generate some core intuitions as to how the crisis came to
be.
The first model, the multi-armed bandit model, explains why opioids were
approved for use. When seeking drug approval, a pharmaceutical company
runs clinical trials to demonstrate drug efficacy and a lack of deleterious
side effects. We can model a clinical trial as a multi-armed bandit problem
where one arm corresponds to prescribing the new drug and the other arm
corresponds to a placebo or the existing drug.

A Model of Opioid Approval
Multi-Armed Bandit Model
To demonstrate their efficacy, opioids were tested against placebos. In
clinical trials, patients were randomly assigned to take either opioids or a
placebo. The assignment of the opioid can be modeled as one arm of a twoarmed bandit and the placebo as the other arm. At the end of treatment,

each trial is classified as a success or a failure. Clinical tests found that
patients who received opioids experienced (statistically) significantly less
pain. Tests on patients who had hip replacements, dental surgery, and
cancer treatments all found that opioids outperform placebos.
With any drug, the potential for addiction is a concern. Tests showed that a
small percentage of patients, fewer than 1%, became addicted, allowing the
drug to be approved. Those tests did not take into account the possibility
that doctors would write longer prescriptions, in some cases a month’s
supply. The longer an individual takes opioids, the more likely that person
becomes addicted. Empirical addiction rates exceeded 2.5% for patients
with longer prescriptions. The Markov model shown in the box below
shows how an increase in those rates from 1% to 2.5% can increase the
equilibrium number of addicts 5-fold.

Transition-to-Addiction Model
Markov Model
A three-state Markov model reveals a nonlinear relationship between
transitions to addiction and overall addiction rates. The model’s states
represent people not in pain, people using opioids, and addicts. We estimate
the transition probabilities between those states, which we represent as
arrows. The model on the left assumes that 1% of people who use opioids
become addicted and that 10% of addicts revert to the no-pain state. It also
assumes that 20% of the people in the no-pain state become opioid users. In
equilibrium, only 2.2% of the population are addicts. To account for longer
prescriptions, the model on the left assumes that 2.5% of people who use
opioids become addicted and that 5% of addicts revert to the no-pain state.
It also assumes that 20% of the people in the no-pain state become opioid
users. Now, in equilibrium, 10% of the population are addicts.1
image
The model’s transition probabilities are only loosely calibrated to data. We
are using the model to build intuition for how a relatively small rate of
addiction can lead to a large number of addicts. By experimenting with the

model, we find that if we lower the probability of leaving the addict state
and increase the probability of moving from the no-pain state to the opioid
state, then the proportion of addicts can increase dramatically. If, for
example, we lower the transition from addict to no pain to 1% in the second
model, the proportion of addicts increases to 35%. One implication of this
type of model thinking has been that some health care providers, such as
Blue Cross, now limit the number of pills a doctor can prescribe. In
addition, some states, including the State of Michigan, have passed laws
restricting the number of pills in a single prescription.
Our third model relies on systems dynamics. This model, like the Markov
model, assumes that there are people in pain, people who use opioids, and
people no longer in pain. Rather than write transition probabilities between
these states, however, it imagines a flow from people in pain to opioid use
to people not in pain. More elaborate systems dynamics models can include
sources for other drug providers, and allow for movements between opioid
and heroin users. In addition, a richer model could include heterogenous
types of potential users. The fact that people who suffer from anxiety and
depression are more likely to become addicted could therefore be included
in the model.2

Paths to Heroin Addiction
Systems Dynamics Model
A population of people in pain produces opioid users and addicts. People
on opioids flow into the no-pain state and also flow into the addict state.
Addicts, in turn, can become heroin users. One reason that people use
heroin is that they can no longer get opioids. Thus, as the flow of opioids
increases, so does the number of heroin users.
image
A possible final model, which we do not write down formally, relies on
social networks to explain why maps of per capita opioid use show
clustering in rural counties. From our analysis of the square root rules, we
know that smaller populations should have higher variation. (Recall the

example of the best and worst performing schools being small.) Higher use
in rural areas could also be due to doctors who write prescriptions for more
pills to accomodate rural patients who live farther from pharmacies. Those
explanations aside, the clustering exceeds what would occur randomly.
Clustering could arise if people provide or sell opioids to neighbors. Unlike
used furniture, which people sell by placing ads, opioids cannot be sold in
the open market. Most often, people sell them through trusted personal
connections. A network model of opioid selling in which people distribute
opioids through family and friends would likely produce local clusters of
opioid abusers. The extent to which those clusters resembled the clusters
seen in the data could be used to draw inferences about who sells to whom.
And, that model could then be used to create interventions.
Other network and social influence models delve into what drives the
choice to take or prescribe opioids or the potential for social influences to
help people overcome their addictions, and even the interactions between
the local economy and opioid addiction. Does, for example, the need to
show up to work increase the prevalence of opioid use or reduce the
likelihood of addiction? Or, perhaps it does both? Such questions, of
course, require gathering data, but what data to gather and how to interpret
it would be informed by models.3

Many Models and the COVID Pandemic
Models played prominent roles in the world’s response to the COVID
pandemic. Fatality rate models communicated the potential impact of the
pandemic. Curve fitting and SIR models forecasted numbers of cases and
locations of outbreaks. Empirical measures derived from models such as
the effective reproduction number guided policy choices and individual
behaviors. Elaborate microsimulation (agent-based) models informed
decisions about what parts of society to close and what to keep open, and
calibrated models of specific sectors and industries informed how reopening occurred. The use of models to frame decisions became so
prevalent during the COVID pandemic, that the terminology of
epidemiological models became part of our daily conversations. We

worked to flatten the curve, to reduce the probability of transmission and
prevent superspreaders.
A comprehensive analysis of the many models that were put to use during
the pandemic—their successes, failures, and impact—could, and surely
will, fill multiple volumes. Here, we focus on some of the more prominent
models and their contributions. To frame how the models were used, some
basic facts about COVID, though widely circulated, bear repeating. In
December of 2019, a deadly strain of coronavirus, creating a disease known
as COVID, was identified in the Wuhan province of China. Lethal variants
of a coronavirus can cause severe acute respiratory syndrome (SARS).
By the end of January, the World Health Organization had declared it a
global pandemic. Coronaviruses, named for their crown-shaped appearance
when viewed under a microscope, are not rare. Along with rhinoviruses,
they are a leading cause of the common cold. These can spread from person
to person when someone coughs or sneezes. Droplets from sneezes can
then linger in the air or land on surfaces and transfer the virus.
While the media focused much of their attention on the use of models to
make forecasts of cases, hospitalizations, and fatalities, by far the more
important contribution of those models was to inform interventions. By
structuring our reasoning, models guided actions and informed the time and
duration of interventions (Jewell, Lewnard, and Jewell 2020).
This dialogue between data and models happened rapidly. Empirical
analyses of interventions based on models helped us understand the spread
of COVID in Italy (Gatto et al. 2020) and on the potential effects of global
travel restrictions (Chinazzi et al. 2020). These models were invaluable in
helping countries and regions make wise policy choices. For example,
almost all models showed that without stringent efforts to slow the virus’s
spread, upward of tens of millions of people would die. As a result,
governments around the world restricted air travel, closed schools, and, in
some cases, quarantined entire populations. Interventions were enacted
broadly, even in countries and regions that had few cases because leaders
trusted science and were persuaded by the evidence that had been
structured by models. Nothing was spared. The 2020 Olympic games,

music festivals and cultural events, and professional conferences were all
canceled or moved online.
In early March 2020, the Centers for Disease Control and Prevention
announced that their models suggested that up to 1.7 million people in the
United States could die from COVID and that the number of
hospitalizations could exceed 20 million. To put those hospitalization
numbers in perspective, the United States has approximately one million
hospital beds (Fink 2020). Other projections were even more dire. A model
developed by researchers from Imperial College in London estimated that
without interventions, 2.2 million people in the United States and over
500,000 people in Britain would die from coronavirus (Ferguson et al.
2005, Ferguson et al. 2020).
These models played two important roles. Calibrated versions of these
models helped to guide the allocation of respirators. Graphical versions that
showed how flattening the curve would prevent hospitals from being
overcrowded educated the public as to the need for drastic measures. To be
effective, laws and policies require public support. Models can help build
public support and drive behavioral changes. Unfortunately, not everyone
believed what the models predicted. As would have been expected, people
who trust science and understand models were more supportive of
interventions than those who do not.
Models were also used to explore questions about the potential effects of
viral mutation and how the efficacy of a vaccine will depend in large part
on whether fear of the vaccine overpowers fear of the disease (Epstein
2020). If an insufficient number of people get vaccinated, and if groups of
people who fear vaccines live near one another, they will be at risk of
creating conditions for outbreaks.
Before discussing specific models, we should take a moment to discuss the
difficulties associated with making accurate predictions about the extent of
a pandemic whether measured in infections, hospitalizations, or fatalities.
The central difficulty is that credible models change the world they are
forecasting. A model that policymakers trust, one that predicts millions of
deaths, often turns out to be a self-unfulfilling prophecy. We want
predictions of dire consequences to be selfunfulfilling! We want

governments and people to take actions to prevent those deaths from
occurring.4
To be accurate in the long run, a forecasting model must therefore predict
the actions that governments will take and people’s responses. Neither of
those is easy. Government actions are notoriously hard to predict, and we
spent an entire chapter on the difficulty of modeling people. Add to these
difficulties the fact that a virus spreads largely through individual contacts,
and as a pandemic begins its spread, our contact networks change.
Predicting how people will change their behavior in a novel situation is
difficult. How many of us will quarantine? Wear masks? These are hard
questions to answer. We should not expect any model to be accurate, and
even if one were, we probably could not identify that model before the fact.
The smarter approach, the one advocated in this book and followed by the
Centers for Disease Control and Prevention, relies on many diverse
models.5 Anthony Fauci, the director of the National Institute of Allergy
and Infectious Diseases, often noted the CDC’s reliance on multiple
models.
To encourage the development of many models, the CDC sponsors annual
contests among models to predict flu incidence. In those contests, the most
accurate models are themselves ensembles of models. In the 2017–2018
influenza season, which by the way was the worst in the past decade with
approximately 45 million people infected, a consortium of four research
teams created an ensemble that combined twenty-one models. They
weighted the models so as to maximize the ensemble’s accuracy (Reich et
al. 2019).6 That ensemble model outperformed all twenty-one of its
component models.
The CDC contests embody the many-to-one principle. They also, as it turns
out, satisfied the one-to-many principle. Detailed microsimulation (agentbased) seasonal flu models could be rewritten as COVID models. The
ability of modelers to retool existing models endowed the CDC with
multiple models to interrogate when making projections about cases,
hospitalizations, and fatalities from COVID. All of those models, of course,
would be wrong. But collectively, they would prove useful, and far more

accurate than claims made without models that the virus would disappear in
summer or lead to a few thousand fatalities at most.
We begin our more formal analysis of COVID models by describing how
fatality rate models were used to estimate approximately 2 million fatalities
without interventions.7 A fatality rate model expresses the expected
number of fatalities as the product of three terms: the population size, the
proportion of people infected, and the fatality rate. To make a prediction,
we need to assign values to each of the three terms on the right side of the
equation.

Fatality Rate Model
The number of fatalities can be expressed as the product of the population
size, the percentage of people infected, and the population level fatality
rate.
Fatalities = Population Size x % Infected x Fatality Rate
No Intervention Estimate:
2,500, 000 = 333, 000, 000 x 30% x 2.5%
Best Case Scenario:
100, 000 = 333, 000, 000 x 3% x 1%
One of those terms is known. The population of the United States is
approximately 330 million people. We must make predictions or forecasts
of the other two terms. Data from China, Italy, and Spain gathered in the
spring of 2020 estimated the fatality rate from COVID as being between
1% and 2.5%. The actual fatality rate for COVID depends on who catches
the virus, which depends on behavior. Would people more at-risk take
greater precautions? Recall that our task here is to predict fatalities
assuming no extraordinary response. We might then assume that people
would respond to the coronavirus in latter stages of the pandemic just as
they had at the beginning. If so, we can conservatively estimate the fatality

rate to be 2.5%. In later, more sophisticated models, we will lower that
estimate.
To estimate the proportion infected, we can look to past pandemics. The
1918 flu had a similar basic reproduction number to the novel coronavirus
and infected approximately one-third of the world’s population. In 1913,
people traveled less and fewer people lived in cities. Medical care was less
advanced, and information flowed much more slowly. These societal
changes point in opposite directions. For more virulent flu viruses, such as
the 2017–2018 seasonal flu, upward of 15% of people show symptoms.
That year 40% of people received flu shots, which reduces the likelihood of
illness by up to one-half. Combining all of this information, the one-third
figure seems a reasonable initial benchmark.
If we multiply those three numbers together, we arrive at 2.5 million
fatalities. To put that number in perspective, COVID loomed a larger threat
to human life than the ten leading causes of mortality—heart disease,
cancer, accidents, respiratory failure, stroke, Alzheimer’s, influenza and
pneumonia, nephritis, and suicide—combined. Actions were necessary.
We can also apply the fatality rate model to construct a best case scenario.
To do so, we again use flu virus data to benchmark estimates. CDC data
show that the flu season with the fewest cases in the past decade was 2011–
2012 in which approximately 3% of the population was infected. That
season’s flu vaccine had an estimated effectiveness of 47%, which is
relatively high. While there would be no available vaccine for COVID in
the first six months, the proposed interventions for COVID could, in a best
case scenario, reduce the incidence of the disease to that of a mildly
virulent flu. For a fatality rate, we might optimistically assume that medical
treatment and isolation of the most vulnerable could reduce the fatality rate
to 1%. In a moment, we will give that assumption more justification.
Multiplying those three numbers together, we arrive at a best case scenario
of 100,000 fatalities, a 20-fold reduction in the number of fatalities.
We can now return to the question of how to estimate the realized fatality
rate, which will depend on how people react to the virus. Recall the three
types of models of human behavior: rational choice, sociological, and
psychological. To make a robust prediction of the fatality rate, we should

consider all three. A rational-choice approach would predict that people
balance the risk of death against the enjoyment of going out to dinner. If so,
high-risk people will be far more likely to quarantine, lowering the realized
fatality rate. That behavioral prediction requires that people know their
risks. General health issues—obesity, diabetes, heart damage, respiratory
problems, and the like—were known comorbidities.
Age functions as a reasonable proxy for these comorbidities. Early data
showed that older people were more likely to suffer fatalities. Figure 29.1
shows the approximate population size of the United States broken up into
ten-year age groups along with rough estimates of the COVID fatality rates
for each group. The youngest two categories each contain approximately 40
million people and have fatality rates of less than one-fifth of a percent.
The oldest two categories together contain 40 million people. These groups
have fatality rates of 8% and 15%—more than forty times the fatality rate
of the youngest categories.8
image
Figure 29.1: Population Sizes and Fatality Rates for US Population by Age
A rational-actor model would assume that people over fifty recognize the
risk and quarantine themselves or, at least, more actively engage in social
distancing. Given that people under fifty suffer few fatalities, a purely selfinterested rational-actor model might assume that they reduce their social
distancing. A rational-actor model that assumes people care about the
health of others (which most people do) would assume that younger people
who do not sequester would choose to avoid older people.
Sociological models of behavior would assume that highly educated
scientifically minded people would place themselves at less risk than
people who are skeptical of science, or who feel immune to ill effects of
COVID. This last group would consist of mostly young people. That model
might even assume that some young people and religious people would
ignore the advice to socially distance and continue to meet in large groups.
Thus, the sociological model would assume, contrary to a rational choice
model, that some high-risk people would choose risky behaviors and pay
the consequences.

Last, psychological models would assume that more affable, conscientious,
and introverted people would be better at sequestering, and that people who
are more neurotic, or who lack self-control or are more extroverted would
be more likely to socialize. These personality characteristics could be
captured in a model by adding personality type to a rational choice
framework, and having personality enter into decisions. The result would
be that some low-risk people would be more likely to sequester and some
high-risk people would not.
If we combine all three of these models, we might expect (optimistically)
that only 1% of people over fifty become infected, but that say 4% of those
under fifty do. The large proportion for the younger people would result
from both rational calculations and sociological conformity. Those
assumptions produce an estimate just shy of 80,000 fatalities, a number
close to our previous estimate of 100,000.9 More sophisticated models such
as the ones we cover later produced similar projections.
The central takeaway is that interventions, if successful, could save
millions of lives. However, there was no guarantee that interventions would
produce numbers as low as the best case scenario. Estimates of how well
interventions might succeed varied. One that informed policy in the United
States was developed by the Institute for Health Metrics, and Evaluation
(IHME). Early on, this model predicted that with successful interventions,
the United States would have only 60,000 fatalities. The model, developed
by Christopher Murray and colleagues, received an enormous amount of
attention in the media and from politicians. Media coverage gave the
impression that the IHME model alone was being used to predict outcomes
and inform policies. As already noted, that was not the case; the CDC
relied on multiple models. Even had the IHME model been known to be by
far the best model (it was not), the CDC would have relied on multiple
models.
The popularity of the IHME model stemmed from its simplicity, its
brashness (the model presented narrow ranges of likely outcomes), and its
exceptional graphical interface. Each of these three attributes positioned the
model for media amplification. Ironically, use of the IHME model spread

through a contagion process. When leading media sources promoted the
model, other media sources did the same.
While the features of the IHME model that drove its contagion—simplicity,
confidence, and graphics—should not be the primary reasons for choosing
among models, each can be justified. All else equal, simpler models are
often better. They reveal logic more clearly and they can be communicated
with less effort. Higher statistical confidence, the narrower outcome ranges,
are also preferred. Who would not want a more accurate model? However,
the error ranges reported by the model were conditional on assumptions
that could not be expected to hold—as we learn when we unpack the
model. Finally, choosing a model based on graphics may sound ridiculous,
but the quality of graphics may signal quality of programmers or level of
funding. The IHME model had deep-pocketed funders, including the Gates
Foundation. As we learned from signaling models, people who lack full
information rely on signals to make choices. In this instance, the Gates
Foundation signal combined with fabulous graphics, functioned as the
equivalent of a peacock’s feathers.
To understand how the original IHME model made its predictions, we need
to recall the SIR model from our study of contagious processes. That model
shows that viruses will produce an S-shaped pattern of the number of
infected people. The initial upward slope results from the increasing
number of infected people. As the number of susceptible people decreases,
the slope of the curve attenuates.
The initial IHME model was not based on an underlying SIR model.
Instead, the model assumed that the number of cases and fatalities could be
approximated by a logistic function. A logistic function is an S-shaped
curve with time as the independent variable.
The shape of the S depends on three parameters: a growth rate, a maximum
value, and a center. To make their predictions, IHME researchers “fit” data
on the number of fatalities to a logistic curve. They used fatalities rather
than the number of cases because early in a pandemic when testing is not
widespread, fatality data are more accurate.

By fitting the data to the logistic function, they could estimate the growth
rate, center, and maximum value, which in this case corresponds to the
number of fatalities. The initial data that they fit to the logistic curve came
from people largely unaware of the virus and an almost complete absence
of government policies to reduce the virus’s spread. Policies to quarantine,
social distance, and close schools and businesses were in the process of
being enacted.
The IHME model had to confront the question of how to model people’s
responses to the interventions. Here, the IHME team ran into a difficulty:
they had no explicit model of behavior, so how could they model a
behavioral response? They had constructed their estimate by leveraging the
fact that diseases produce S-shaped curves and used data to fit such a curve.
That approach is less a “model” than an exercise in fitting data to a curve
derived from a model.
To account for the effects of government policies, they fit a second curve.
Using data from regions in China, Spain, and Italy that had imposed strong
stay-at-home orders and reduced the spread of the virus, they constructed a
second curve. IHME now had two fitted curves. The first was a United
States curve, which was the sum of a collection of state-level curves
estimated from data. That curve had a high maximum number of fatalities.
Call this the no-intervention curve. The second curve, which had a much
lower maximum number of fatalities, corresponded to an effective
interventions curve.

The IHME Model
The IHME model assumes that the empirical number of fatalities (cases)
closely fits a logistic function.
image
This function produces an S-shaped curve as a function of time, t, and three
parameters: a maximum value, m, a growth rate, g, and a center, c.
image

Using data on the number of fatalities, IHME estimated the parameters for
each state in the United States. Using post-intervention data from China,
Italy, and Spain, they also estimated parameters assuming interventions, a
post-intervention estimate. If states adopted intervention policies, their
estimated logistic function was predicted to shift to the post-intervention
estimate.
The IHME team assumed that interventions would move the United States
curve toward the effective interventions curve. To make their estimates,
they assumed that each of four policies, school closings, closings of
nonessential businesses, stay-at-home orders, and travel restrictions, would
shift the US curve one-third of the distance to the China-Italy curve, with a
maximum shift of the full amount. In other words, they assumed that once
three of the policies had been adopted, that adding a fourth policy had no
effect. That assumption could not be correct, but it built in conservatism on
the effect of interventions.
Their assumptions may seem ad hoc. To an extent, they are. Why should
each policy have the same effect? Why would the fourth policy have no
effect? And, perhaps most troubling, why would the effects be additive? It
is not clear that any one policy would be that effective alone. Those
critiques aside, these were not unreasonable assumptions about policy
efficacy. They just turned out to be wrong. People did not behave as
(implicitly) predicted.
A deeper problem with the initial IHME model, or any curve-fitting
exercise, is that it presumes that the spread of the virus will follow an Scurve. The model cannot predict a second wave in which a second S-shaped
curve, a second wave, occurs. Second waves arise when few people are
infected initially, and society then opens up, allowing the virus to spread
anew. This occurred with the 1918 flu virus, and second waves of the
coronavirus also appeared in late summer of 2020.
The “cannot get a second wave if you fit a curve” criticism may be a bit
harsh. Think of the IHME model as a short-term model to predict where
and approximately how fast the virus was likely to spread. For those
purposes, the model was useful. (Yes, in the long run, the criticism holds.)
To predict a second wave, the model would have to make explicit

assumptions about numbers of infected and susceptible people and rates of
transmission. The IHME team knew this, and in later versions of their
model, they included an SIR model as part of their calibration.
Before describing how to build a coronavirus-specific SIR model, we first
return to the basic SIR model to show how it was used in communicating
risk and in evaluating policy efficacy. Recall that with the SIR model, we
can calculate the basic reproduction number, R0, which equals the product
of the probability of spreading the virus and the probability of contact
divided by the probability of recovery. If R0 exceeds one, the virus will
spread, and if it is less than one, the virus will dissipate. Early estimates of
R0 for the coronavirus were around 2.5, a number that when combined with
the fatality rate caused sufficient concern for actions to be taken.
Those interventions and the resulting changes in behaviors changed the
probability of spreading and the probability of contact. Epidemiologists,
therefore, calculate the effective reproduction number, Rt, which is a
function of time, t.
One quick and powerful method for gauging the success of interventions is
to track Rt by region. Estimated effective reproduction numbers can
identify hot spots and guide policy at regional and state levels. If estimates
of Rt far exceeded one in a region, policymakers knew to take more
aggressive actions to slow the spread of the virus. The often heard phase of
“flattening the curve” refers to reducing the effective reproduction number.
According to the SIR model, the goal of policy interventions during a
pandemic should be to reduce the effective reproduction number below
one. The difference between an effective reproduction number of 1.1 and
an effective reproduction number of 0.9 equates to the difference between
the disease continuing to spread through the population and the spread
slowing.

Flattening (and Unflattening) the Curve

In the SIR model, the number of newly infected depends on the effective
reproduction number. Reducing that number through more social
distancing and fewer contacts flattens the curve. It reduces the maximal
number of new cases, which occurs at a later date.
image
Flattening the curve reduces the peak number of cases. In doing so, it
improves the capability of a health care system to respond. We do not need
a model to know that. What the model can tell us, when calibrated to data,
is approximately how much lowering the effective reproduction number
will reduce the maximum number of beds needed. The SIR model also tells
us something less intuitive: flattening the curve pushes back the peak date.
This gives health care systems more time to prepare. In brief, lowering the
effective reproduction number offers a win-win. It reduces the maximal
number of cases and delays when those occur.
The policies that closed businesses and schools and limited travel proved
successful, at least initially. As shown in figure 29.2, in Michigan, the
effective reproduction number fell well below one soon after interventions
were imposed. Over time though, people became less vigilant.
Quarantining became difficult. People wanted to socialize. As evident in
the graph, when policies were relaxed, the effective reproduction number
crept back above one, meaning that the virus had picked up steam. In
Michigan, by August, the effective reproduction number would once again
fall below one. Figures for most other states followed a similar pattern,
with the effective reproduction number falling below one but then
increasing. That pattern reveals the adaptive response to the threat. When
people perceived risk, they acted in such a way to reduce the virus’s spread.
When they felt the virus was under control, people relaxed, allowing the
virus to spread more quickly. We saw that same dynamic in our ping-pong
model covered earlier in the book: when people take actions that oppose a
trend, they reverse the trend.
image
Figure 29.2: Sequestration Policy Effects on Rt in Michigan (rt.live)

Many of the models that government officials and health administrators
used for making short-term forecasts of the number of cases and
hospitalizations relied on variants of the SIR model. These models were
calibrated to COVID and included features like a latency stage, degrees of
infection, and the possibility of hospitalizations. The latency period, the
time in which a person has the virus but shows no symptoms, could range
from two to five days depending on the individual. Someone who
experienced symptoms might recover on their own or require
hospitalization.
We will describe one such model built by Marissa Eisenberg, a
mathematical epidemiologist. Rather than three states—susceptible,
infected, and recovered—the model includes eight states connected by ten
arrows. Each arrow corresponds to a probability or rate of moving from one
state to the next. The values assigned to those arrows are calibrated by
region to develop regional policies.
In deciding how large to make the regions, the modeler once again
confronts the bias-variance tradeoff. Make the regions too large, say an
entire state, and the model’s predictions will be biased for regions that
differ demographically or in their density of connections. Make the regions
too small, and the lack of data leads to high variance in predictions.

SIR with Latency and Severity (Eisenberg)
At any moment in time, each member of the population can be classified in
one of eight states. The circle on the far left represents people who are
susceptible, S. Once infected, people enter a latent (L) state where they
cannot infect other people. After exiting the latent state, individuals either
recover (R), or become infected. Infections can be either severe (Is) or mild
(Im). In either of the infected states, people could infect other people.
People with severe cases become hospitalized (H). After entering the
hospital, they either recover or suffer a fatality (F). People with mild cases
do not enter the hospital. They either recover or seek care (C) and then
recover.

image
A model that bases its assumptions on a large population, such as the entire
state of Michigan or New York, would underestimate the fatality rate for
regions with more older people and might underestimate infection rates for
more dense regions. For instance, people living in upstate New York and
the Upper Peninsula of Michigan are older but interact with fewer people
than people living in New York City and Detroit. So, you would not want to
lump upstate New York with New York City or the Upper Peninsula with
Detroit.
Models like this do more than make predictions. They help policymakers
reason. The Eisenberg model included a dashboard that had two sliders.
Adjustments to the first slider changed how much people reduced their
contacts.10 Contact rates could be set to some percentage of normal. The
second slider set a start date and an end date for when social distancing
occurred. According to the model, if no social distancing occurred,
Michigan would have had over 20,000 cases per day in mid-April. If social
distancing were put in place from April 1 until May 31 and if people
reduced their contacts to 25% of normal, then in mid-April, the peak
number of cases would have been cut in half.
The model also revealed the potential for a second wave if the interventions
were strict and imposed for too short a time period. According to the
model, if people reduced their contacts to 10% of normal for two months
and then society returned to normal levels of contact, a second wave would
occur in which nearly as many people would catch the virus in July and
August as in April and May. This potential for a second wave weighed
heavily on the minds of policymakers.
Keep in mind that given the difficulty of making accurate long-term
predictions of a first wave, we should be skeptical of any model’s
quantitative estimates of the likelihood of a second wave or its size. And
yet, every ounce of skepticism about numerical estimates should be
balanced by a pound of engagement with the logic the model reveals: too
much flattening of the curve could leave so many people susceptible and
invite a second wave. We would expect that the first signs of a second wave

would lead to more shutdowns. That in fact happened in many states:
policies that shut down parts of the society were reinstated. From our pingpong model analogy, we should have expected interventions to reappear,
and, more generally, should have known beforehand that managing the
pandemic would require constant vigilance.
More detailed SIR models like the one just covered do not take full
advantage of the types of data now available. The next model we cover, a
microsimulation (agent-based) model developed by researchers at Imperial
College in London, does. It can be best understood as a giant simulation of
the real world (Ferguson et al. 2020). Agents in the computer model
represent individual people who live in communities, belong to families,
and go to work and school. Using census data, the model matches
household sizes and age distributions. The model includes data on
workplace sizes (not firm sizes), lengths of commutes, and school
populations. The model distributes schools based on population density.
Denser regions have more schools.
The Imperial College model was not constructed anew. It was a
repurposing of an earlier model of the flu virus (Ferguson et al. 2005). To
apply that model to COVID, the researchers performed the same type of
calibration as in the previous cases. They estimated latency and varying
levels of infectiousness. Because the model differentiates between where
someone catches the virus, it can (and does) assume different rates of social
contact. Based on past flu data, schools were assumed to have double the
per capita contacts as the workplace, home, or community. That calibration
implied that without interventions, infections would be equally likely to
occur at work or school, within households, or in neighborhoods.

Imperial College Microsimulation Model
The model assumes four locations in which the virus could spread:
neighborhoods, households, schools, and workplaces. Agents
representing individuals in the population live in households (Hl) that are
parts of neighborhoods (Ni). Household sizes and demographics and
neighborhood densities and demographics are calibrated to data. Younger

agents within a household attend schools (Sk) in their neighborhoods with
school and classroom sizes calibrated to data, meaning that neighborhoods
with higher population density have more schools. Where people work (Wj)
is assigned to calibrate with commute times.
image
The arrows in this diagram correspond to agents from neighborhoods going
to school and work. Within each household and location, the model
assumes an SIR model with latency and severity of infectiousness and a
basic reproduction number of 2.4.
The Imperial College model includes far more detail than the other models.
By including so much detail, the model enables us to explore what happens
if schools close but workplaces are left open, or to estimate the
consequences of keeping schools open while shutting down workplaces and
limiting community interactions.
Models like this one that allow for many types of people and locations can
also include heterogeneous behaviors. Some people may elect to follow the
rules and maintain social distance, wear masks, and not gather in large
groups. Other people may not. By altering behavioral assumptions within
the model and by exploring the effects of policy alternatives, policymakers
can identify crucial communities, behaviors, and policies. The estimates
will not be perfectly accurate. If they are off by less than a factor of two or
three, they will be of great value. Without a model, we would be left to
conjecture the effects. Our intuitions about dynamical processes are so poor
that we might make decisions that cost many lives.
The models we have covered so far have considered regions or entire
countries. Industries, sectors, and organizations also built models to inform
designs for reopening. We consider here one such model developed at
Cornell University that used enrollment data to evaluate risk. Kim Weeden
and Ben Cornwell constructed this model to calculate how many students
co-enrolled in classes. They considered the entire university,
undergraduates, and liberal arts majors. All three co-enrollment networks
could be described as small world networks. Such networks have short path

lengths. Short path lengths facilitate sharing ideas, so universities like short
path lengths. Unfortunately, so does a virus.
The numerical estimates from Cornell shocked college administrators. On
average, each Cornell student took at least one class with 2.4% of the
student body. Unfortunately, each student was also only one person
removed from 60% of the population, and two students removed from over
95% of the population (Weeden and Cornwell 2020). In other words, start
with any student at Cornell and you can get to any other student through
just two other students. Viruses thrive on networks such as these.
To make sense of how these numbers could arise, we can perform the same
sort of algebraic calculations used to explain six degrees of separation
between any two people. Cornell University has approximately 15,000
undergraduates. Assume a student enrolls in one large class with 200
students, two medium-sized classes with 60 students, and two small classes
with 20 students. That student would be co-enrolled with 360 students.
With no overlap in students, that would amount to 2.4% of Cornell’s
student body. If, on average, those 360 students each took classes with only
25 other unique students, then the number of unique students who are
connected to our initial student through just one other student would be
9,000, or 60% of the student population.11 Each of those 9,000 students
would need, on average, fewer than one unique connection to reach the
95% proportion for students one more connection away.
Based on the Cornell findings, other universities built models of their entire
student population to explore different procedures for opening their
campuses. Those exploratory models argued strongly against holding large
classes. They led to design ideas for creating cohorts or bubbles of students.
Similar applications of models can be found in every sector. As should be
evident from this cursory overview, in helping society respond to the
coronavirus, models performed all seven uses. They helped us reason and
explain patterns. They were used to communicate concepts. They made
predictions that guided actions and informed the design of interventions.
The models helped administrators explore the implications of reopening
plans.

On a final note, a great deal of ink has been spent characterizing how
wrong the models have been. The IHME model with its estimates of 60,000
and then 80,000 fatalities has been a particular focus of criticism. Those
projections implicitly assumed that Americans would lock down as well as
people in other countries. We did not. The model proved wrong. Other
models though, including simple fatality rate models, were nearly accurate.
Imagine for a moment that we had not had models. How much worse might
the responses of countries around the world have been? How many
resources would have been misallocated? How many lives would have been
unnecessarily lost? Or, imagine if everyone had believed the models and
acted accordingly. How many lives might have been saved?

Many Models of Inequality
Our final many-model exercise delves relatively deep and wide into the
causes of economic inequality. We undertake this effort for three reasons.
First, inequality is one of the most important policy issues of our time.
Income and wealth correlate with human flourishing. Higher-income
individuals enjoy better health, longer life expectancy, and higher life
satisfaction and happiness. Those at the bottom of the income distribution
experience higher rates of homicide, divorce, mental illness, and anxiety.12
We must be careful not to confuse correlation with causation: a substantial
part of this correlation can be explained by the fact that healthier, happier
people earn more money. Nevertheless, almost all studies show a
connection between income and flourishing. No one prefers to be poorer.
Second, we have a plethora of models of inequality written by a diversely
tooled collection of economists, sociologists, political scientists, and even
physicists and biologists. Third, we have abundant data on income and
wealth within and across countries. We have both current data and time
series stretching back hundreds of years.
We start by summarizing some empirical regularities. First, in all countries
at all times, the distribution of income has an elongated tail, with many
low-income people and a small percentage of people who earn large
incomes. Historically, income distributions were calibrated to lognormal
distribution or Pareto distributions. Recently, more granular data reveal the

tail to be longer than lognormal, though not quite that described by a power
law. Wealth distribution is similarly skewed.
image
Figure 29.3: Income Shares of the Top 0.1%, 1916–2010. Source: Piketty
2014.
Second, within most developed countries, income and wealth inequality,
however measured, have been rising in recent decades. Current levels of
income and wealth inequality in the United States approach those of the
Gilded Age. Shifts in entire distributions can be hard to discern, so,
following convention, we describe those shifts with respect to the share of
income that goes to the upper tail of the distribution. Figure 29.3 shows
how the top 0.1% has increased its share of income. The share of income to
the top 0.1% of families fell steadily through 1950, and remained stable at
less than 4% until around 1980, when it began to climb. In 2018, the
proportion of total wealth owned by these super rich was around 10%.
Third, globally, the number of people living in poverty has dropped
precipitously. We should see no logical contradiction in these opposing
trends. Fast-rising incomes in poor countries reduce cross-country
differences and more than offset within-country increases in inequality. Our
model of group selection produced similar effects. The growth in the
number of altruistic communities outpaced the trend toward selfishness
within each community.
Inequality has multiple, interwoven causes. Economic forces, sociological
trends, exercises of political power, and the weight of history all contribute
to disparities. Thus, as Steven Durlauf points out, we should not try to
explain the levels or trends in disparities with a single equation. Nor should
we base policy on one.13 We must be nuanced in our thinking. The
processes concentrating wealth and income in the top 1% or top 0.1% may
be unrelated to the forces trapping the bottom 20% in a cycle of poverty. To
understand the disparate causes of inequality requires a variety of models.
We start by describing models that explain the changing distribution of
income. Income has four sources: wages and salaries, business income,

capital income, and capital gains. The relative sizes of those shares vary by
income level. Low-income people earn few capital gains or capital income.
Many of the highest earners receive substantial income from every
category. They earn income from wages, businesses, and capital.
Our first model extends the Cobb-Douglas production model to include two
types of labor: educated and uneducated. The wage paid to a type of labor
depends on the relative supply of that type and on technology.14 This model
explains the recent rise in inequality based on supply and demand.

Technology and Human Capital Model
Growth Model
Output depends on physical capital (K), educated labor (S), and
uneducated labor (U) as follows:
Output = AKαSβ Uγ
The parameters A, α, β, and γ capture the technology and the relative value
of the three types of labor. The relative market wage for high- and lowskilled workers is as follows:15
image
Cause of inequality: Technological changes that favor educated workers
increase β and decrease γ. These changes, along with increases in the
supply of low-skilled workers, increase inequality.
During the 1950s, the rise in manufacturing increased demand for
uneducated workers. At the same time, increased college enrollments due
in part to the GI Bill increased the supply of educated workers. In the
1980s, decreased incentives to attend college slowed the growth in the
number of college graduates, and a subsequent inflow of immigrants with
low education levels increased the supply of low-skilled workers. At the
same time, technological changes—the rise of automated manufacturing
and the transition to a more digital economy—increased the relative value

of educated workers, and their rising wages reflected this change in relative
value.
Time series data on average incomes by education level fit this model
reasonably well. For this reason, many economists rely on the model to
guide policy. The model advocates increasing access to education, as that
will depress the wages of educated workers and reduce inequality. This
model explains broad trends well, but it cannot explain the increase in
variation within each income class.
The next model, the positive feedback model, can explain the increased
variation within professions. It focuses on the tail of the distribution and, in
particular, on entrepreneurs. In 2011, entrepreneurs made up 70% of the
400 wealthiest individuals in the United States.16 The model assumes that
technologies—the internet and smartphones in particular—have made us
more connected and more influenced by the choices of others.17 A person
buying wireless stereo speakers can read reviews online and select “the
best” from among a dozen choices. In the past, that person might have had
a single option at her local stereo store. Now, a person who twists her knee
can search the web and learn the identity of her favorite athlete’s doctor.
That behavior creates a positive feedback and more inequality. We model
socially influenced economic choices by reframing the preferential
attachment model as a model that links positive feedbacks to talent.

Positive Feedbacks to Talent
Preferential Attachment Model
There exist N producers, and each begins with zero sales. The first
consumer buys from a random producer with zero sales, giving that
producer positive sales. Each subsequent consumer with probability p buys
from a producer with zero sales, and with probability (1 − p) buys from a
producer with positive sales. When buying from a producer with positive
sales, a customer selects randomly, with the probability of choosing a
particular producer that is proportional to that producer’s sales.

Cause of inequality: Increased connectedness increases social influences,
creating a positive feedback.
Though the positive feedbacks model cannot be fitted to time series data
with the same fidelity as the previous technology model, we can look to
experiments to see how feedbacks contribute to inequality. Recall the music
lab experiments described in Chapter 6. College students sampled and
downloaded music under two treatments. In the first treatment, subjects
could not see what music others had downloaded. This treatment captures
the pre-internet world. In the second treatment, subjects could see the
download numbers for each song. In the treatment without social
information, no song receives more than two hundred downloads and only
one song receives fewer than thirty. When people can see downloads, one
song receives more than three hundred downloads and over half receive
fewer than thirty. Information and social influence amplify the Matthew
effect. The rich get even richer, and the poor become relatively poorer.
We can apply that same logic to the economy writ large.18 The potential for
positive feedbacks through social networks to contribute to inequality
depends in part on the nature of what people buy. Weightless goods such as
movie and music downloads, web applications, and some technologies can
be scaled quickly, if not immediately. Tractors, cars, and washing machines
cannot be duplicated by clicking on an icon. So, while a new smartphone
application can scale up with little to no capital outlay, a best-selling car
cannot. As a benchmark, in May 2015, Volvo announced that it would build
its S60 sedan in South Carolina. The company broke ground on the plant
three months later. In 2018, the first S60s rolled off the line. Due to
changing demand, the plant will also produce sport utility vehicles, but not
until 2022.
Our next model applies the spatial voting model to explain the rise in CEO
pay, which is not determined by social forces. In 2012, the average income
of a CEO at a Fortune 500 company exceeded $10 million, or roughly 300
times the average pay of a worker. By comparison, in 1966, the CEO made
only about 25 times the average worker’s salary. CEOs in other countries
earn much less. In Japan, CEOs earn about 10 times what the average

worker does. In Canada and throughout Europe, CEOs earn approximately
20 times the pay of the average worker.
At most large companies in the United States, the CEO’s pay is set by a
compensation committee consisting of members of the board of directors.
That pay includes salary, bonuses, and stock options. The people who
determine the pay of CEOs are often other CEOs. They have an incentive
for the pay of other CEOs to be high in order to drive up their own pay. We
can use the spatial model to represent the preferences of the compensation
committee. According to the spatial model, the salary will be set at the
median voter’s preferences. The difference in CEO pay by country can be
explained by the composition of boards and the compensation committees.
In Germany, boards of directors include workers, who prefer that the CEO
be paid less.
The model explains the rise in CEO pay based on board members
preferences over CEO. The preferences of compensation committee
members could be informed by models or data. Or, those preferences might
also be socially constructed or even part of an elaborate log roll, in which
CEOs vote to raise one another’s pay.

CEO Political Capture
Spatial Voting Model
CEO pay is determined by a vote of a compensation committee. In the
United States, compensation committees include many current and former
CEOs, who prefer higher pay, as well as compensation experts (X). Other
countries include workers (W) on compensation committees, resulting in a
median voter who prefers much lower pay.
image
Cause of inequality: CEOs determine their own pay through capture.
Increases in the pay of any one CEO shifts preferences toward higher pay
for all CEOs.

Our next model of income inequality comes from Thomas Piketty’s bestselling book Capital in the Twenty-First Century. Picketty’s analysis relies
on models that show that the rate of return on capital exceeds the growth
rate of capital. When that holds, the portion of income that high-income
individuals receive from returns on capital will increase over time. By
constructing more elaborate versions of the growth models from Chapter 8,
it can be shown that the return to capital will generally exceed the rate of
growth in the broader economy. Over the long haul, an economy might
grow at less than 2% or 3%, but returns to capital will be greater.
It follows that in an economy that consists of workers who earn wages and
capitalists who earn income from rents, the share of income going to
capitalists will increase relative to the share that goes to income. To be a bit
more formal, the rate at which capital increases will depend on three rates:
the consumption rate, the tax rate, and the return on capital.
Consumption depends on the level of capital. A person with little capital
will consume a large percentage of her income. A person who owns a
substantial amount of capital will consume a small percentage of her
income. As shown formally in the box below, if we make the consumption
level constant, the consumption rate will equal that amount divided by the
level of capital. Thus, wealthier people will consume at a lower rate making
it more likely that their net capital increases.

Rent-from-Capital Model (Piketty)
Rule of 72
The economy consists of workers and capitalists. The wages of workers
increase at a rate g, the growth rate in the economy. The capitalists have
wealth Wt at time t and earn return r (net of taxes) and consume a constant
amount A. The income of capitalists will increase faster than that of
workers if and only if
image

Cause of inequality: In a market economy, the rate of return on capital
exceeds the overall rate of growth (r > g). Capitalists with large
accumulations of wealth spend a small proportion of their income from
capital on consumption, so their share of total income increases over time.
To see how the difference in rates produces inequality, we can apply the
rule of 72. If initially the incomes of workers equal those of capitalists and
wages grow at 2% while capital grows at 6%, then in thirty-six years wages
double but income from capital increases 8-fold. Within seventy-two years,
capitalists earn sixty-four times the income of workers.
Piketty applies this model to explain long-term trends in inequality of both
income and wealth. The model calibrates remarkably well with three
centuries of data from France and England. The model also sheds light on
patterns of inequality over the past century in the United States and Europe,
in which the two world wars destroyed capital stocks in Europe, evening
out the income and capital distributions there. One reason the model fits the
data as well as it does is that it omits two effects that cancel out. By
excluding entrepreneurs, the model understates inequality. In assuming that
all succeeding generations of capitalists invest wisely—not all do—the
model overstates capital accumulation’s contribution to inequality. The
creation of a new class of rich individuals and the loss of an old class of
rich individuals need not balance out. A more granular model would
include movement in and out of the wealthy class.
That caveat aside, the model’s implication is that so long as capital
increases, capitalists earn an increasing portion of the economic pie. If we
keep applying the rule of 72, we find that the income of the capitalists
eventually dwarfs that of the workers. The problem of capital accumulation
has a straightforward solution: impose a wealth tax. That may not be
politically possible. As an alternative, we might wait for a war or revolution
to redistribute wealth by force or for technological breakthroughs that
produce a new set of wealthy capitalists.
Our next two models give priority to sociological forces. Both also have
strong empirical support. The first explains rising inequality based on
assortative mating. A family’s income depends on the incomes of both
partners. If a low-income person marries a high-income person, then that

marriage will contribute toward equalizing income distributions. If highincome people marry other high earners, then income disparities will
increase. Most people marry at an age when a potential partner’s lifetime
income cannot be known with certainty. People do know the education
level and general health of potential partners and get signals of their
ambitions. Evidence shows that as men and women become more educated
and earn higher incomes—refer back to the technology and human capital
model—they choose life partners who also have higher education levels.

Assortative Mating
Sorting Model and Categories
Each individual has an education level: {1, 2, 3, 4, 5} where 1 = dropout, 2
= high school diploma, 3 = some college, 4 = college degree, and 5 =
postgraduate.
Let P(m, j) and P(w, j) denote the probability that a man and woman have
education level j. Income (g, l) equals the (estimated) income of a person of
gender g and income level l. Household income for a couple consisting of a
man with education level lM and a woman with education level lW earns the
following estimated household income:19
Income(M, lM) + Income(W, lW)
Cause of inequality: Increases in the number of educated women,
increased pay for workers with higher levels of education, and assortative
mating (the tendency for people to marry others of the same income level)
result in an increase in household-level income inequality.
The increase in inequality results from the following factors. First, women
increasingly earn college degrees. Second, relative income increases with
education level. Third, educated men and women prefer educated partners.
Therefore, families with two educated people will be more likely to have
two high incomes contributing to household-level income inequality. The
logic is airtight. The only question concerns the size of the effect.20

Sociologists calibrate the model by categorizing people into five education
levels: dropout, high school, some college, college degree, and
postgraduate. They then calculate the average income for each education
level and fit the data for the number of marriages between each pair of
education levels, resulting in a crude approximation of the impact of
assortative mating.
Had marriages been random rather than assortative, income inequality
would be much less. One study finds that inequality as measured by the
Gini coefficient, a common measure of inequality, would have decreased
by 25%.21
Our next model analyzes movements between income categories using a
Markov model. It categorizes people (or households) by income level: high,
upper middle, lower middle, and low. Each category contains one-fourth of
the distribution. Given a time period—it could be a year, a decade, or a
generation—we can then estimate the transition probabilities between
income categories to capture mobility.
If there were no stickiness across generations, then the income of the child
of a high-income parent would be equally likely to belong to any of the
four income classes—all of the transition probabilities equal image. In
the most extreme case of no mobility, transition probabilities would consist
of only 1s along the diagonal. Empirical estimates suggest that the reality
lies between these extremes.
We can run experiments by taking 100 randomly selected low- or highincome families and computing the probability distribution of incomes in
subsequent generations. Using the probabilities shown in the box, the
children of high-income parents have a 60% chance of being high-income
and only a 5% chance of being low-income. The grandchildren of the highincome parents have less than a 43% chance of being high-income and
more than a 10% chance of being low-income.22
The income dynamics model also serves as a baseline from which to
evaluate the causes of income mobility. We might use a linear model to
estimate a child’s income as a function of parental wealth, parental income,

and parental ability levels (assuming we had data). The Piketty model
would imply a positive coefficient on parental wealth. The ability-based
model would imply a positive coefficient on parental ability given that
there exists some correlation between parental ability and ability of
offspring.

Intergeneration Income (Wealth) Dynamics
Markov Model
The population can be divided into four income (or wealth) categories with
equal numbers of people. We can estimate the transition probability that
an individual (or family) in one category moves to another category within
a generation, as shown in the figure below. More equal transition
probabilities correspond to greater social mobility.
image
Transition Probabilities Between Income Levels
Cause of inequality: Social skills, tacit knowledge, attitudes toward risk
and education, and bequests reduce mobility between income classes.
Note that determining the coefficient on parental income requires data on
the income of each child and each parent. Scholars have individual-level
income data only for the past few decades. In The Son Also Rises, Gregory
Clark (2014) found a novel solution to the problem of lack of data: he relies
on surnames. He calculates the average income of everyone named, say,
Thatcher, in 1888 and compares this to the average income of everyone
named Thatcher in 1917. The thirty-year increment represents the length of
a work life. He finds substantial correlation across surnames’ average
incomes, suggesting a lack of income mobility.
This type of model allows us to identify racial differences in
intergenerational transfers. African Americans exhibit less persistence of
wealth at the top of the income distribution and more persistence at the low
end. A wealthy African American will be less likely to have wealthy

children, and a poor African American will be more likely to have poor
children.23
Our last model based on neighborhood effects, Durlauf’s persistent
inequality model, leverages the empirical regularity that people segregate
by income category—that is, high-income people live in communities with
other high-income people and low-income people live near low-income
people. Segregation by income produces economic, sociological, and
psychological externalities that reduce mobility. In the model, an
individual’s income depends on ability, educational spending, and
spillovers.

Persistent Inequality (Durlauf)
Schelling Segregation Model + Local Majority Model
Individuals belong to income classes and segregate residentially by income.
Individuals allocate a portion of their income to education, resulting in
positive spillovers that increase with community income level. The future
income of a child living in community C depends on her innate ability,
spending on education, and spillovers. The contributions of education and
spillovers depend on the level of income within the community, IC.
IncomeC = F(ability, education(IC), spillover(IC))
Cause of inequality: Children who grow up in low-income neighborhoods
receive fewer educational opportunities and economic spillovers.
The educational attribute captures public spending on education, which
empirically correlates with average income: high-income locations spend
more on education than low-income locations, resulting in better
educational outcomes and higher incomes for children in high-income
neighborhoods.
The spillover term can be interpreted as socially transmitted knowledge of
appropriate tools to acquire. Here we can link Durlauf’s model to how
people who live in high-income communities gain awareness of appropriate

tools. We can also link the model to our network model and the strength of
weak ties phenomenon: people who live in high-income communities will
be connected indirectly to more people with access to economically
valuable information. This will produce a positive feedback on income.
We can also interpret the spillover as socially transmitted behaviors, such
as the number of hours spent studying or working. If income includes a
random component, then a person in a low-income community will see
(correctly) a low return to time spent on self-improvement. Relatedly, the
spillover could include psychological attributes—a positive or negative
outlook on life, a feeling of safety, or a belief in oneself.
In the complete model, Durlauf solves for equilibrium levels of educational
spending and derives conditions in which persistent inequality arises. That
inequality results from what he calls poverty traps. Individuals living in
low-income communities lack the educational resources and levels of
spillovers necessary to earn high incomes regardless of their ability levels.
Durlauf’s model can help to explain the enormous racial gaps in income
levels. African Americans disproportionately live in poor neighborhoods,
and as a result, they may become trapped in low-income trajectories owing
to a lack of resources on multiple dimensions.
Sociologists refer to this as compounded disadvantage (Sampson 2019).
The persistent inequality model calls into question the efficacy of what we
earlier defined as big coefficient thinking to reduce racial inequality.
African Americans attend weaker schools, have access to fewer family
resources—both in terms of financial capital and human capital. They live
in neighborhoods with more crime and less healthy water and air.
Identifying which one of these impediments has the largest effect and
fixing it will have little impact if the other forces remain in play. Such
efforts would be like small steps up a slippery hill, quickly erased by a
larger force. This model suggests that to be effective, a policy must work to
create a new reality, which would require multiple simultaneous large
actions.
We have now covered a number of models that each describe a distinct
cause of income inequality. In a sense, each is correct, but, as we know,
each model is also wrong. This can be seen by examining their explanatory

contributions. The models vary in how much and what part of the variation
in income they can explain. For the upper end of the income distribution,
the empirical evidence most strongly supports the models that rely on
technological change.24 For over twenty years, the IRS has tracked the
highest 400 incomes. Those at the top of the distribution come from
technology, mass retail, and finance, three industries that can scale quickly.
That high growth rate could stem from winner-take-all markets for search
engines (Google) or social networking sites (Facebook). These models tell
us little about the lower end of the income distribution. Nor do they say
much about income mobility, or explain why CEO pay in the United States
far exceeds that in other countries.
To explain these other features of the data, we need the other models, such
as the income mobility model, Durlauf’s persistent inequality model, and
the spatial voting model. By constructing a dialogue between multiple
models and data, we come away with a deep, multifaceted understanding of
the causes of inequality. We identify multiple processes that produce and
maintain inequality and see how they overlap and intersect. Our
understanding of the complexity of inequality and the self-reinforcing
causal forces that sustain it should make us dubious of quick fixes. As
discussed in the context of the persistent inequality model, real change will
require concentrated efforts on multiple fronts.

Into the World
We have just learned how by applying many models as an ensemble we can
explicate the multiple causes of the opioid epidemic and income equality
and reveal the limits of any one model to predict outcomes of a pandemic.
Were we policymakers, we could fit some of these models to data to gauge
effect sizes. We could then run experiments to help us guide policy choices
based on what we have learned. Findings from experiments of limited
duration and scope, though informative, should be interpreted through
multiple models. For example, systemic effects or feedbacks may reverse
or mitigate the effects found were the experiment to be expanded.
The core theme of this book, that we need many models, to reason, act,
design, and so on, can be applied to any number of our present challenges.

Reversing trends in obesity, improving school performance, mitigating
climate change, managing water resources, and improving international
relations can all benefit from a many-model approach.
Adding even a single new model can have enormous consequences. Take
for example the problem of predicting financial collapses. The United
States Federal Reserve relies on traditional economic models using national
accounting data on inflation, unemployment, and inventories. Those data
suffer from lags. They are released weekly, quarterly, or annually. Those
data also come from surveys, that is, samples of the entire economy.
Complexity scholar J. Doyne Farmer argues for creating a second class of
models based on real-time data scraped from the web. These new models
would rely on more granular, instantaneous data, and therefore differ from
traditional models. Farmer argues that such models could prove much
better than existing models. He may be right. Yet, these new models need
not be more accurate to be of use in predicting and preventing financial
disasters. Given the new models would use different data and rely on
different assumptions, so long as they are not far less accurate, when
combined with existing models, these new models would improve
predictions. Policymakers, to use Farmer’s turn of phrase, would become
more collectively aware.25
Business leaders and policymakers might engage in a similar exercise when
making important decisions. They could apply multiple models informed
by data to decide on product or policy attributes, time product or program
launches, design compensation plans, construct supply chains, and forecast
sales or impact. Each of these actions occurs within a complex system. No
single model will suffice to excel at any of those tasks. Many models will
be needed.
That logic applies when confronted with any choice—when asked to make
a prediction, or when faced with a design challenge, we should take a
many-model approach. Many-model thinking produces better performance
than taking actions based on hunches and gut instincts. That said, we have
no guarantee of success. Even with many models, we may not identify the
most relevant logical chain. The domain of interest might be so complex

that even ensembles of models can only explain a small portion of the
variation.
The same holds when applying models to aid in design; we may find
ourselves unable to construct useful abstractions. The simplicity of models
may, in those cases, be their undoing. In the face of complexity, it is
possible that we find models not up to the tasks of communicating ideas,
making accurate predictions, or pointing us toward the best actions.
Our explorations with models may often lead us down rabbit holes. We
might not be able to identify a new policy or a behavioral intervention
likely to improve the world. Nevertheless, even in those cases, we benefit
from contemplating and applying models. And, we benefit all the more
from applying many models as they will clarify our thinking by uncovering
interdependencies.
Complex processes, such as those that result in epidemics and inequalities,
often frustrate our attempts to understand, explain, or communicate. That is
not a reason to give up on models and to rely on our intuition, but a reason
to press on, to continue to develop new and better models.
As we learn to apply those models, we must maintain a degree of humility.
Even when using many models, our abilities to reason have limits. By
definition, complex phenomena are difficult to predict, explain, and
understand. We must keep our minds open. We must continue to build new
models and to improve upon existing ones. If a model leaves out key
features of the world—such as social influences, positive feedbacks, or
cognitive biases—then we should build other models that include those
features. By doing so, we can begin to discern when those attributes matter
and how much. The fact that all models are wrong should not take the wind
out of our sails but instead motivate us to develop more models. With many
models, we have the possibility of wisdom.
We should also seek joy in our modeling efforts. Throughout the book,
much of the focus has been on pragmatic ends—to become better thinkers,
to be more effective at work, and to operate as more informed citizens of
the world. We have, in a few spots, taken notice of the beauty and elegance
of models. When we see model building as an art as well as a science, we

begin to recognize the beauty of the exercise. We make the assumptions,
write the rules, and then allow our minds to play within those rules. The
laws of logic outline the boundaries and we explore within those limits.
Through logical play, we improve ourselves and become wiser. May we
take that wisdom out into the world and help to change it in positive ways.