8. Concavity and Convexity
To say nonlinear science is akin to saying non-elephant zoology.
—John von Neumann
We now introduce nonlinear models and nonlinear functions. Nonlinear
functions can curve downward or upward, they can form S-shapes, they can
kink, jump, and squiggle. In time, we cover all of these possibilities. We
start here with models that rely on convexity and concavity. We show how
growth and positive feedbacks produce convexity and how diminishing
returns and negative feedbacks produce concavity. Most disciplines contain
models of both types. Economic production models assume that delivery
and inventory costs decrease with a firm’s size, making profits per unit sold
a convex function of a firm’s size, which explains why Walmart earns such
large profits.1 Economic models of consumption assume that the utility (or
value) is concave, that we enjoy the fifth piece of pizza less than the first.
In ecosystems, when a new species invades and confronts no predators, its
population grows at a constant rate, producing a convex function. As that
population grows, it has less food. Fitness, as a function of population size,
is therefore concave.
The chapter consists of three parts. The first part covers models of
population growth and decay. The second part covers concavity. In it, we
see how concavity implies risk aversion and a preference for variety. In the
third part, we study a series of growth models from economics that
combines concave functions and linear functions.
Convexity
Convex functions have an increasing slope: the function’s value increases
by a larger amount as we increase a variable’s value. The number of
possible pairs of people is a convex function of the group size. A group of
three people includes three unique pairs. A group of four people includes
six unique pairs, and a group of five includes ten unique pairs. Each
increase in group size increases the number of pairs by a larger amount.
Similarly, each time a chef adds a new spice to his repertoire, he increases
the number of spice combinations by a larger amount.
Our first model of convexity, the exponential growth model, describes the
amount of a variable, often a population or a resource, as a function of its
initial value, a growth rate, and the number of periods.
Exponential Growth Model
A value of a resource at time t, Vt, that has an initial value of V0 and grows
at a rate R can be written as follows:
Vt = V0(1 + R)t
This single-equation model plays central roles in finance, economics,
demography, ecology, and technology. When applied to finance, the
variable is money. Using the equation, we can calculate that a $1,000 bond
paying 5% annual interest increases in value by $50 in year one and by
more than $100 in year twenty. To draw clean inferences, we assume a
constant growth rate. Given that assumption, we can manipulate the
exponential growth equation to derive the rule of 72.
Rule of 72
If a variable grows by a percentage R (less than 15%) each period, then the
following provides a good approximation:
Periods to Double ≈
image
The rule of 72 quantifies the cumulative effect of higher growth rates. In
1966, Zimbabwe had a per capita GDP of $2,000, twice that of Botswana.
Over the next thirty-six years, Zimbabwe experienced little growth.
Botswana, meanwhile, averaged 6% growth, meaning that Botswana’s
GDP doubled every twelve years. In thirty-six years, it doubled three times,
an 8-fold increase. Thus, in 2004, Botswana’s per capita GDP of $8,000
was four times that of Zimbabwe.
This same formula reveals why housing bubbles must end and
technological progress need not. In 2002, home prices in the United States
rose by 10%. That would imply a doubling every seven years. Had that
trend continued for thirty-five more years, prices would have doubled five
times—a 32-fold increase. A house costing $200,000 in 2002 would cost
$6.4 million in 2037. Prices cannot rise at that rate. The bubble had to
burst. In contrast, Moore’s law states that the number of transistors that can
fit on an integrated circuit doubles every two years. Moore’s law has
persisted because spending on research and development has generated a
near constant rate of improvement.
Demographers apply the exponential growth model to human populations.
A population that grows at 6% a year doubles in size in twelve years. In
thirty-six years, it doubles three times, and in one hundred years, it doubles
eight times (increasing 256-fold). In 1798 British economist Thomas
Malthus noticed that the population was growing exponentially and wrote a
model showing that if the economy’s ability to produce food only increased
linearly, then a crisis loomed. The short version goes as follows: Population
was growing like 1, 2, 4, 8, 16, 32,.… Food production was growing like 1,
2, 3, 4, 5,.… Malthus foresaw disaster. Fortunately, birth rates fell, and the
arrival of the Industrial Revolution increased productivity. Had nothing
changed, Malthus would have been correct. But he ignored the potential for
innovation—the focus of models later in this chapter. Innovation subverted
the trend.
The exponential growth model can be applied to the growth of species as
well, and not just to rabbits. When you acquire a bacterial infection, tiny
bacteria reproduce at incredible rates. Bacteria in human sinuses grow at
around 4% a minute. By applying the rule of 72, we can calculate they
double every twenty minutes. In a single day, each initial bacterial cell
spawns over a billion offspring.2 Their growth stops when the physical
constraint of your sinuses leaves them no room. Food constraints,
predators, and lack of space all reduce growth. Some species, such as deer
in suburban America or the hippos brought to Colombia by drug lord Pablo
Escobar, encounter few constraints on growth and their population grows
rapidly, though not at bacterial rates.3
A convex function with a positive slope increases at an increasing value. A
convex function with a negative slope becomes less steep. A convex
function with an initially large negative slope will flatten. That is true for
the equation in the half-life model, which captures decomposition,
depreciation, and forgetting.
In the model, every H periods half of the quantity decays. Hence, H is
known as the half-life for that process. For some physical processes, the
half-life is constant. All organic matter contains two forms of carbon: an
unstable isotope, carbon-14, and a stable isotope, carbon-12. In living
organic matter, these isotopes are present in a constant ratio. When an
organism dies, the carbon-14 in its body starts to decompose with a half-life
of 5,734 years. The amount of carbon-12, on the other hand, does not
change. Willard Libby, a physical chemist, realized that by measuring the
ratio of carbon-14 to carbon-12, one can estimate the age of a fossil or
artifact, a technique known as radiocarbon dating. Paleontologists apply
radiocarbon dating to the remains of dinosaurs, woolly mammoths, and
prehistoric fish. Archeologists use it to adjudicate claims of authenticity.
The remains of Ötzi the Iceman, discovered in the Italian Alps, were
estimated to be five thousand years old. The Shroud of Turin, first
displayed in 1357 and claimed to be Christ’s burial shroud, was found to
date from the fourteenth century and not the time of Christ.
Half-Life Model
If every H periods half of the remaining quantity decays, then after t
periods the following holds:
Proportion Remaining ≈
image
A novel application of the half-life model comes from psychology. Early
psychological studies showed that people forget information at a nearconstant rate. Our half-life of remembering depends on the salience of the
event.4 In 2016, the film Spotlight won the Academy Award for Best
Picture. If people’s memory of Oscar winners has a half-life of two years,
in 2018, image of people will have remembered that fact, but by 2026,
only image will recall it. The recollection of any particular event varies
across people. Tom McCarthy, who directed and cowrote Spotlight, will
likely never forget the year he won the Academy Award.
Concave Functions
Concave functions are the opposite of convex functions. Concave functions
have slopes that decrease. Concave functions with positive slopes exhibit
diminishing returns: the added value of each extra thing diminishes as we
have more of that thing. Our utility or value from almost all goods exhibits
diminishing returns. The more leisure, money, ice cream, or even time
spent with loved ones, the less we value having more of it. Evidence for
this can be found in the fact that the more we consume of just about
anything, including chocolate, the less we enjoy it and the less we are
willing to pay for it.5
Diminishing returns can explain a variety of phenomena, including why
long-distance relationships are often so happy. If you see your partner just a
few hours each month, every additional minute is wonderful. After a month
of uninterrupted togetherness, the slope of the happiness curve flattens, and
those few extra moments matter less.6 It explains why developers invite
people for free weekend visits to their beachfront condominiums. During a
short weekend, you cannot get enough time on the beach. You are inclined
to buy. After ten days on the beach, though, you may become bored.
When we assume concavity, we imply a preference for diversity and risk
aversion. To show the former requires a concave function with multiple
arguments. If our happiness is concave and increasing in both leisure and
money, we prefer some leisure and some money to all leisure and no money
or all money and no leisure. Risk aversion means a preference for a sure
thing over a lottery. A risk-averse person prefers a certain payoff of $100 to
a lottery that pays $200 half of the time and nothing the other half of the
time. A risk-averse person prefers a double-dip ice cream cone to having
either no ice cream or an unwieldy four-scooper.
Figure 8.1 shows why concavity implies risk aversion. The figure plots
happiness for values for three outcomes: a high outcome (H), a low
outcome (L), and the mean of those outcomes (M). Given the downwardshaped curve, happiness at the mean outcome exceeds the average
happiness of the low outcome and the high outcome. The opposite holds for
convex functions. Convexity implies risk-loving: we prefer the extremes to
the average. The amount of a stock you can buy is a convex function of its
price. Therefore, buyers of stocks prefer price volatility. If prices go up and
down, buyers end up with more stocks than if prices stay constant.7
Economic Growth Models
We next construct a series of economic growth models. These models
reveal the causes of growth and can explain and predict growth patterns
across countries. They can also guide actions such as increasing the savings
rate. To lay the foundation for our study of growth models, we introduce a
standard economic production model in which output depends on labor and
physical capital. Empirical evidence and logic support concavity of output
in both labor and capital. Holding the amount of capital fixed, labor should
be worth less as more is added. Similarly, adding more machines or
computers adds less value given a fixed number of workers. Logic also
suggests that output should be linear in scale. Doubling both the number of
workers and the amount of capital should double output. A broom-making
company with sixty workers and one factory that builds a second factory
and hires sixty additional workers should double its output. The CobbDouglas model, one of the most widely used models in economics, includes
both properties. Output is concave in labor and capital and linear in scale.
This model can be applied to capture production by single firm or by an
entire economy.8
image
Figure 8.1: Risk Aversion: Value(Mean) > Mean of the Values
Cobb-Douglas Model
Given L workers and K units capital, the total output equals:
Output = Constant · La K(1−a)
where a is a real number between 0 and 1 capturing the relative importance
of labor.
We use the Cobb-Douglas model to construct models of economic growth.
To simplify, we assume 10,000 workers in the economy and ignore wages
and prices, allowing us to focus on how the number of machines affects
total output. We can then connect investment in capital to growth. To make
the model as simple as possible, we assume that output takes the form of a
single commodity, coconuts. The coconuts provide flesh and rich milk for
food. However, the coconuts grow high in trees, so the workers require
machines to pick them. We then make the very unrealistic assumption that
the machines are constructed from coconuts. This simplifies the model but
maintains the key trade-off between consumption today and investment in
the future As a special case of the Cobb-Douglas model, we write output as
the square root of the number of workers times the square root of the
number of machines.
image
If the economy has one machine, output equals 100 tons. If people consume
all 100 tons of coconuts, they invest in no new machines. Output will be
unchanged in the next year. The economy exhibits no growth. If they invest
1 ton of coconuts to build a second machine, output increases to 141 tons, a
41% growth rate. If they build a third machine, output grows to 173 tons.9
Through a constant investment, the economy grows at a decreasing rate.
Output is a concave function.
Simple Growth Model
Production Function: O(t) = 100 image
Investment Rule: I(t) = s · O(t)
Consumption-Investment Equation: O(t) = C(t) + I(t)
Investment-Depreciation Equation: M(t + 1) = M(t) + I(t) − d · M(t)
O(t) = output, M(t) = machines, I(t) = investment, C(t) = consumption, s =
savings rate, and d = depreciation rate
Now that we have the basic idea of how investment drives growth, we can
construct a more elaborate model that includes an investment rule. We can
write investment as savings rate times output and assume a fixed
depreciation rate on the machines, such as that the number of machines
that are no longer useful at the end of the year equals a fixed proportion of
the number of machines. We can then write the total number or machines in
the next year as last year’s machines plus the investment in new machines
minus the machines lost to depreciation. The complete simple growth
model consists of four equations.
If we assume the economy has 100 machines, a savings rate of 20%, and a
depreciation rate of 10%, output equals 1,000 tons of coconuts,
consumption equals 800 tons, and new investment equals 200 machines. A
total of 10 machines will be lost to depreciation, leaving 290 machines at
the start of the new year. Similar calculations show that in the second year,
outcome will equal 1,702 tons and in the third year it will equal almost
2,500 tons.10 In the first three years, output increases at an increasing rate.
This initial convexity is a result of the small number of machines in the first
few years implies almost no effect of depreciation. Over time the number of
machines grows and depreciation starts to matter making output concave.
In the long run it ceases altogether, as shown in figure 8.2. By analyzing the
model we can see why. Investment is linear in output: the number of new
machines added grows linearly with output. Output is concave in the
number of machines, so as the economy grows, investment will also be
concave in the number of machines. Depreciation, though, is linear in the
number of machines, and eventually the linear depreciation catches up with
the concave increases in production.
image
Figure 8.2: Output in the Basic Growth Model for One Hundred Years
In the long-run equilibrium of the economy the number of new machines
created by investment equals the number lost to depreciation. In our model,
the equilibrium occurs when economy has 40,000 machines and produces
20,000 tons of coconuts. At that point, the economy invests 20% or 4,000
coconuts, in new machines and loses exactly that many machines to
depreciation (10% of the 40,000). Thus, the number of new machines lost
to depreciation equals the number of new machines created through
investment and growth stops.11
The Solow* Growth Model
We now construct a more general model that is a simplification of the
Solow growth model (thus the asterisk). We replace machines with physical
capital and include labor as a variable. We also add a technology parameter
that increases output linearly. Innovations increase this parameter. As in the
previous model, the long-run equilibrium occurs when investment equals
depreciation. Here, though, the equilibrium-level output depends on the
amount of labor and on the technology parameter, as well as the savings
and depreciation rates.12
Solow* Growth Model
Total output in the economy is given by the following equation:
Output = A image
where L denotes the amount of labor, K denotes the amount of physical
capital, and A represents the level of technology. The long-run equilibrium
output, O∗, is given by the equation13
image
Long-run equilibrium output increases in the amount of labor, the growth in
technology, and the growth in the savings rate. It decreases with a rise in
the depreciation rate. None of these results is surprising. More workers,
better technology, and more savings increases output, and faster
depreciation reduces output. The fact that output increases linearly with
labor and savings is less intuitive. Labor produces diminishing returns, so
without working through the model, we might expect long-run output to be
concave in the amount of labor. However, as the amount of labor increases,
so too does output, which in turn increases investment, leading to more
output. The positive feedback from investment exactly offsets the
decreasing returns. Last, equilibrium output is convex with the depreciation
rate. Lowering the depreciation by 20% increases output by 25%.
Finally, long-run equilibrium output increases as the square of the
technological improvements. Innovation therefore increases output more
than linearly. We can use the model to show why. If we start with an
economy in a long-run equilibrium and increase the technology parameter
by 50%, output increases by 50%, and so too does investment. Investment
then exceeds depreciation, so the economy continues to grow. Investment
continues to outpace depreciation until the economy has grown another
50%, at which point the capital lost to depreciation offsets investment.
These calculations reveal that innovation has two effects, creating an
innovation multiplier. First, innovations directly increase outputs. Second,
they indirectly lead to more capital investments creating an additional
increase in output. Innovation, therefore, is the key to sustaining growth.14
These increases in output do not occur instantaneously. When a
breakthrough occurs, the technology parameter changes slowly. The direct
effects unfold over time. Old physical capital must be replaced by new
physical capital with the better technology. A company’s computers do not
get faster when technology changes; they get faster when technology
changes and the company buys new computers. The second-order increase
that results from the increased investment in physical capital takes place
over an even longer time frame. Lags between technology and its effects on
growth can imply that an innovation produces growth over a period of
decades. Trains were invented in the early 1800s. The Gilded Age did not
begin until the latter part of that century, a gap of over fifty years. The
internet boom took place three decades after the creation of the
ARPANET.15
Why Nations Succeed and Fail
We can apply our growth models to big policy questions such as whether
backward countries can catch up, why some countries succeed and some
fail, and the role of government in promoting growth. Those investigations
show the value and limits of our models. We can begin with the ability of
low-GDP countries to achieve fast growth. The models show that building
up capital can produce fast growth, as will investing in technology. A
backward country with less physical capital that could jump to the
technological frontier with new capital outlays could experience incredible
growth.16
The necessity of innovation for long-term growth, as shown in the second
model, implies the limits of one-time imports of new technology.
Continued growth requires innovation. Thus, when the Soviets dismantled
German factories and rebuilt them in the Soviet Union following World
War II, they could produce short-term growth, so much so that on
November 18, 1956, Soviet premier Nikita Khrushchev, speaking at the
Polish embassy in Moscow to ambassadors from Western nations,
proclaimed, “Mi vas pokhoronim!,” or “We will be present at your
funeral!” They did not. They failed to do so because the Soviet Union did
not innovate.17 They limited freedom and stifled entrepreneurs.
The models also show how extraction and corruption, the taking of output
from the economy for government use, will reduce growth through reduced
savings. Cross-country comparisons of growth rates support both findings:
reducing extraction and corruption and promoting innovation enhance
growth. Achieving those aims requires a strong but limited central
government that promotes pluralism. The strong center establishes property
rights and rule of law. Pluralism prevents capture by the elite, who often
prefer the status quo and may not embrace innovation, which can be
destructive.
As an example of destructive innovation, consider the website Craigslist,
which posts for-sale and help-wanted ads. In the early 2000s, Craigslist
contributed to the loss of hundreds of thousands of newspapers jobs in the
United States. At that time, Craigslist itself employed only a few dozen
workers. Though people lost jobs, Craigslist made the economy more
efficient by increasing the technology parameter. In a less pluralistic
society, the newspaper industry might have lobbied the government to stop
Craigslist. Doing so would have slowed growth.
Japanese Chinese Economic Dominance
Linear model + rule of 72: From 1960 to 1970 Japan’s GDP grew at a
10% annual rate. A linear projection of continued 10% increases would
result in a doubling of the Japanese economy every seven years (using the
rule of 72). In 1970, Japanese per capita GDP was approximately $2,000 in
current US dollars. Had that trend continued, by 2012 per capita GDP
would have doubled six times, resulting in a per capita GDP of $128,000.
Growth model: This model explains Japanese growth as due to
investments in physical capital. The model predicts concave growth rates
over time. The growth model predicts that as Japan’s GDP approached that
of the United States and Europe, its growth rate should decrease to the
historical cross-country average of 1–2%.18 The evidence supports this.
From 1970 to 1990 Japan’s GDP grew at around 4% annually. From 1990
to 2017, it grew at 1% or less.
Chinese growth: China’s GDP grew at nearly a 10% rate from 1990 to
2010. In 2016, the per capita GDP in China reached approximately $8,000,
and as predicted by the growth model, growth has slowed, with GDP
growing at closer to 6% from 2013 to 2017. In China as well, sustained
10% growth rates run afoul of the rule of 72. If Chinese economic growth
averaged 10% for the next century, per capita GDP would exceed $100
million.
It’s a Nonlinear World After All
We construct nonlinear models because few phenomena of interest are
linear. In this chapter we saw how diminishing and increasing returns are
common features of economic, physical, biological, and social phenomena.
We also saw some of the implications of including curvature in our models.
Most important, perhaps, we saw how functional forms structure our
thinking and then how fitting functional forms to data allows us to make
precise statements. Scientists can compute the age of artifacts using carbon14 data. Economists can estimate the long-term effects of small increases in
growth.
A central takeaway from this chapter is that intuition becomes insufficient
once we include nonlinearities. Intuition tells us the direction of effects:
growth is increased by a rise in savings, an increase in labor, and
technological innovation. Models reveal the shape and form of those
effects. Savings, as we would expect, have a linear effect. Increases in labor
do as well in the long run, even though the model assumes short-run
diminishing returns. Increases in innovation produce a multiplier effect: we
get the square of those effects. The first increase is the direct effect of the
innovation. The second increase in output arises from the increase in
capital.
Insights such as these become clear with the help of models. Without
models, we can usually infer what goes up and what goes down, but we
lack understanding of the shape of functional relationships. As a result, we
often make linear extrapolations—China’s economy will soon take over the
world. With models, we can better think through the logic that produces
nonlinear effects. The set of nonlinear functions is enormous. The concave
and convex models we covered in this chapter represent but a small dip in
that vast sea. If we hope to improve our capacity to reason, explain, and act
in a complex world, we need an even deeper dive into nonlinear
phenomena.