Endogeneity of savings (chapter 3 de unit 2)

The most crucial factor in the Harrod–Domar model is the savings rate, a key
parameter that plays a significant role in economic growth. Whether this parameter
can be easily manipulated by policy depends on the level of control the policy maker
has over the economy. Various factors, such as per capita income and its
distribution, can influence the savings rate. At low income levels, people might not
save much, and in some cases, they might even borrow to meet basic needs.

In societies where most individuals are close to subsistence levels, like in many poor
countries, achieving a high savings rate is challenging. In such situations, external
sources like credit or aid might be essential for capital accumulation and economic
growth. As economies move beyond subsistence levels, there is more room for
savings, although actual savings growth is not guaranteed. Wealthier countries,
despite their high income, may have low savings rates due to changing societal
notions of what is necessary.

The dynamics of savings are influenced by factors like income distribution within a
country and potentially across countries. In richer societies, middle-class aspirations
for prestige and status in the global economy can drive higher savings. The poor,
however, are often unable to save, even if they share similar aspirations. As
countries transition from being very poor to middle-income, there is a tendency for
the savings rate to rise. However, at even higher income levels, the relationship
becomes more ambiguous. While the rich can afford high savings, their relative
position may lead to a decline in the savings rate as current consumption becomes
more attractive.

This perspective requires an adjustment in the Harrod–Domar theory. As incomes

change, the savings rate incorporated into the Harrod–Domar formula will also
change. This implies that the growth rate of a country will tend to alter over time in
alignment with the movement of the savings rate with income. In summary, the
Harrod–Domar model, in its basic form, is a neutral theory of economic growth.
However, when considering the influence of income levels on growth rates, this
neutrality is lost, creating a pattern linking per capita income to growth rates.
This adjustment in the Harrod–Domar theory, while not drastic, introduces a crucial
concept: as incomes change, the savings rate, a key element in the Harrod–Domar
formula, undergoes transformations. This suggests a tendency for a country's growth
rate to evolve in tandem with changes in the savings rate associated with income
levels. For example, following this reasoning, both low- and high-income countries
might experience lower growth rates compared to middle-income countries.

This example reveals an important aspect of the Harrod–Domar model. Initially, it is

a neutral theory of economic growth, providing no inherent explanation for
systematic differences in growth rates at various income levels. However, with the
introduced adjustment, neutrality is compromised. A pattern emerges, linking per
capita income to growth rates, marking a departure from the original neutral stance.
This feature, involving the relationship between income levels and growth rates, will
reappear and be further explored in subsequent discussions in this chapter.
In essence, the Harrod–Domar model, with this adjustment, acknowledges that
economic growth is not universally neutral across all income levels. Instead, it
introduces the idea that the savings rate, influenced by income changes, can impact
a country's growth trajectory. This nuanced perspective enriches our understanding
of the interplay between savings, income levels, and economic growth, revealing a
more dynamic and context-dependent aspect of the Harrod–Domar framework.

The endogeneity of population growth

Just as the savings rate can change with per capita income, population
growth rates also vary. Chapter 9 extensively discusses the significant
evidence suggesting that population growth rates systematically change
with a society's overall development level. This introduces another factor
affecting per capita growth rates independently of the savings rate.

Known as the demographic transition, this phenomenon is crucial in

understanding economic development. In poorer countries, high death
rates, particularly among children, result from factors like famine,
undernutrition, and disease. Consequently, birth rates are high as
families aim to ensure a certain number of surviving offspring. With
improvements in living standards, death rates decline, causing a rapid
initial increase in population growth rates. Over time, as birth rates
adjust, the population growth rate decreases, forming an "inverse-U"
shaped curve known as the demographic transition.

Considering the implications for per capita economic growth, a graph in

Figure 3.2 depicts two lines. The first, representing overall growth (net of
depreciation) in the Harrod–Domar model, remains flat, assuming other
parameters are unaffected by per capita income. The second curve
represents the population growth rate, initially rising with per capita
income and later falling, following the demographic transition.

The growth rate of per capita income, as derived from the Harrod–
Domar equation, is the vertical distance between the two curves at every
per capita income level. This reformulation shows that the rate of per
capita income growth depends on the current income level. The graph
illustrates an "inverse-U" shaped behavior, indicating that per capita
income may rise, then fall, and rise again as per capita income
increases.

The implications for per capita economic growth become clear when
considering critical levels of per capita income marked as "Trap" and
"Threshold." Below the "Trap," per capita income increases over time.
Between the "Trap" and "Threshold," population growth outpaces
income growth, making the economy poorer in per capita terms. Above
the "Threshold," per capita income increases over time, signaling
sustained growth. Without policies pushing the economy beyond the
threshold, it tends to be caught in the trap.

This simplified model suggests that temporary changes in economic

parameters, such as a boost in the savings rate or family planning
initiatives, can have lasting effects. For instance, a jump in the savings
rate can shift overall income growth to outpace population growth,
pulling down the threshold. This allows for sustained growth even if the
policy is not permanent.

The discussion emphasizes the importance of recognizing that factors

considered exogenous, like savings or population growth, may be
influenced by the outcomes they supposedly cause. Understanding
endogeneity in economic parameters can fundamentally alter our
perspectives on the economy and policy. The next model developed by
Solow explores the endogeneity of another parameter in the Harrod–
Domar model: the capital–output ratio.

The Solow model Introduction

Solow's take on the Harrod–Domar model focuses on the law of
diminishing returns to factors of production. This means that when
you have a lot of labor compared to capital, even a small amount of
capital can be very effective in producing output. On the other
hand, when there's not enough labor, the economy tends to use
capital-intensive methods, and the efficiency of additional capital
decreases. This aligns with what we discussed earlier. According to
Solow, the capital–output ratio (represented by θ) is not fixed but
depends on the overall balance of capital and labor in the economy.

To understand the changes in this model, let's go through some

steps similar to what we did for the Harrod–Domar model. We'll
keep equations (3.3) and (3.4) – which are about savings equaling
investment and capital accumulation – without any problem. Also,
assuming that total savings S(t) is a constant fraction s of total
income Y(t), we can combine (3.3) and (3.4) to get an equation.

If we divide everything by the population (Pt) and assume that the

population grows at a constant rate, changing (3.8) to (3.9), where
the lowercase ks and ys represent per capita magnitudes (K/P and
Y/P, respectively).
Before moving forward, make sure you grasp the economic idea
behind the math in (3.9). It's quite straightforward. On the right side,
you have two parts – depreciated per capita capital (which is (1 −
δ)k(t)) and current per capita savings (which is sy(t)). These
combined should give us the new per capita capital stock k(t + 1),
but there's one catch: population growth affects it. That's why the
left side of (3.9) includes the rate of population growth (n). The
faster the population grows, the lower the per capita capital stock
in the next period.

To fully understand the Solow model, we need to connect per

capita output at each time to the per capita capital stock using the
production function. In this model, capital and labor collaborate to
produce total output. With constant returns to scale, we can use
the production function to relate per capita output to per capita
input.

In Figure 3.3, you see a typical production function with

diminishing returns to per capita capital. As per capita capital
increases, the output–capital ratio falls due to a relative shortage of
labor. Keep in mind that output per person continues to rise; it's
just that with a relative shortage of labor, the ratio of output to
capital used in production falls.

Figure 3.4 uses this production function to determine what the per
capita capital stock must be at date t + 1 if the current per capita
stock is k. You can translate equation (3.9) into the diagram by
multiplying the output from any given capital stock by s, which
gives us fresh investment, and adding the result to the depreciated
capital stock. The curved line in Figure 3.4, resembling the
production function, shows this process. It also plots the left side
of (3.9), the straight line (1 + n)k, as k changes. Due to diminishing
returns, the curved line initially lies above this straight line and
then falls below it.
The steady state
Let's simplify what we've learned from this diagram. In Figure 3.4, there are
two scenarios, one starting with a "low" per capita capital stock (Figure
3.4a) and another with a "high" per capita capital stock (Figure 3.4b), both
beginning in 1996.

In the case of the low stock (Figure 3.4a), the output–capital ratio is high
initially, allowing the per capita capital stock to grow rapidly. However, due
to population growth, this growth slows down over time, and per capita
capital eventually stabilizes around a level called k*.

For the high initial capital stock (Figure 3.4b), the per capita stock
decreases over time, converging to the same k* as in Figure 3.4a. Here,
the output-capital ratio is low, resulting in slower expansion of aggregate
capital. Population growth surpasses the growth of capital, leading to a
decline in per capita capital stock.

We can think of k* as a steady-state level where per capita capital stock

settles, regardless of the initial level. This means that growth in the Solow
model slows down if capital is growing too fast relative to labor. Diminishing
returns to capital cause a downward movement in the capital–output ratio,
making the long-run capital–labor ratio constant (represented by k*).

Here's the key point: in the Solow model, if the per capita capital stock
settles at a steady-state level (like k*), per capita income also stabilizes.
Unlike the Harrod–Domar model, the savings rate has no long-run impact
on the growth rate in the Solow model.

Things might seem confusing, especially if we compare it to the Harrod–

Domar model, where the savings rate did affect growth. The Solow model
introduces diminishing returns to capital, which leads to endogenous
changes in the capital–output ratio, affecting long-term growth. Look at
Figure 3.4 again, and notice that the degree of diminishing returns
determines how quickly the per capita capital stock settles down. The
smaller the degree of diminishing returns, the longer it takes for the per
capita capital stock to stabilize—k* becomes larger. The Harrod–Domar
model, on the other hand, assumes no diminishing returns and allows for
indefinite per capita capital stock growth.

In the end, whether the Solow model or the Harrod–Domar model is more
accurate depends on empirical evidence. It's essential to recognize that the
different predictions arise from different assumptions, particularly regarding
technology. As long as we understand these distinctions, there's no need to
be confused.

How parameters affect the steady state

In the Solow model, the rate of savings doesn't impact the long-run
growth rate of per capita income (which is currently zero), but it does
influence the long-run level of income. This is also true for the rate of
depreciation and the population growth rate. These effects operate
through changes in the steady-state level of capital per capita, which, in
turn, affects the steady-state level of per capita output (essentially, per
capita income in the long run).

To express this more formally, consider that if the economy were to start
from the steady-state level of k*, it would remain at k* in every period (as
the term "steady state" implies). Thus, in equation (3.9), we can
substitute k(t) = k(t + 1) = k*. Using y* to represent the per capita output
achievable from k*, we can rearrange terms in (3.9) to derive the steady-
state equation:

s⋅y∗=(n+δ)⋅k∗

Now, let's understand the economic implications behind this equation.

An increase in the savings rate (s) raises the right-hand side of the
equation, requiring an increase in the left-hand side to maintain balance.
This implies that the new steady-state capital–output ratio must be
higher. Considering diminishing returns, this can only happen if the new
steady-state level of k* (and y*) is also higher. Therefore, an increase in
the savings rate raises the long-run level of per capita income.

By the same reasoning, an increase in the population growth rate or the

rate of depreciation will lower the long-run level of per capita income. For
example, a higher rate of depreciation means more national savings
must go into replacing worn-out capital. This results in a smaller net
accumulation of per capita capital, reducing the steady-state level.
Similar logic applies to changes in the savings rate and population
growth rate\

Level effects and growth effects

The population growth rate has a unique impact—it influences both the steady-state
level of per capita income and the overall growth of total income. Higher population
growth reduces the steady-state per capita income, but at the same time, total
income must grow faster. This dual effect stems from the nature of labor, which
serves as both a production input and a consumer of final goods. While the former
effect boosts total output and income growth, the latter lowers savings and
investment, leading to a drop in the steady-state per capita income.

This feature of population growth is interesting because it demonstrates a parameter

that affects income both in terms of its level and its growth. In more technical terms,
we can distinguish between level effects and growth effects. Level effects involve
shifting the entire path of a variable over time, while growth effects change the rate
of growth of the variable. Figure 3.5 illustrates this difference using growth paths of
the logarithm of income over time.

For example, savings rates in the Solow model only have level effects. An increase in
the savings rate raises overall output, leading to a higher per capita capital stock and,
consequently, a higher long-run level of income. However, this doesn't affect the
long-run growth rate; income continues to grow at the rate of population. The
increase in savings has a short and medium-term impact, pushing the economy to a
higher trajectory, but diminishing returns ultimately nullify this effect.

It's important to note that whether a parameter has level or growth effects depends
on the specific economic model. In the Solow model, savings rates have level effects,
but in a world with constant returns to scale, they can have growth effects, as seen in
the Harrod–Domar equation.

. Unconditional convergence
The core idea of the Solow model is the concept of convergence, and there
are different types of convergence. The strongest form is called
unconditional convergence. In simple terms, this means that in the long
run, all countries will tend to have similar values for factors like technical
progress, savings, population growth, and capital depreciation. According
to the Solow model, this leads to a prediction that, regardless of their
starting points, all countries will converge to a common value for capital per
efficiency unit of labor.

This might sound straightforward, assuming similar parameters should

naturally lead to convergence. However, it's not as obvious as it seems.
The claim of unconditional convergence is significant because it suggests
that historical differences in initial conditions, even if countries start at
different levels of per capita income or capital stock, don't matter in the long
run.

Figure 3.7 visually represents unconditional convergence, using the

logarithm of income over time. The line AB shows the time path of per
capita income at the steady state, while the paths CD and EF represent
countries starting below and above the steady state, respectively.
According to the model, these paths will adjust over time, converging
toward the steady-state line AB.

In simpler terms, unconditional convergence means that there is a strong

negative relationship between the growth rates of per capita income and
the initial levels of per capita income. However, this hypothesis is bold, as it
not only assumes countries converge to their own steady states but also
claims that all steady states are identical. Let's now examine the real-world
data to see if it supports this idea.

. Unconditional convergence: Evidence or lack thereof

When testing a hypothesis like unconditional convergence, the first

challenge is dealing with time constraints. Data collection in developing
economies began relatively recently, making it difficult to find reliable
data spanning a century or more. Researchers face two options: either
study a small number of countries over a long time or study a large
number of countries over a shorter period. Let's explore examples of
both approaches.
A small set of countries over a long time horizon

In the study conducted by Baumol (1986), the growth rates of sixteen economically
prosperous countries, which are among the wealthiest today, were examined.
Utilizing data available for the year 1870, thanks to the work of Maddison (1982,
1991), Baumol plotted the per capita income of these countries in 1870 against the
growth rate of per capita income over the period 1870–1979. The objective was to
assess whether the observed relationship aligns with the unconditional convergence
hypothesis, where countries, regardless of their initial conditions, converge to a
common steady state over time.

The exercise seemed to yield favorable results, with the data points approximately
forming a downward-sloping line, supporting the notion of unconditional
convergence. However, there is a crucial statistical caution in this approach. The
countries chosen for analysis were already rich in 1979, introducing hindsight bias.
For instance, Japan, included in the study, was not exceptionally wealthy in 1870. This
raises questions about the validity of the "convergence" found, suggesting it might
be a statistical regularity rather than a true tendency toward convergence.

In response to this concern, De Long (1988) expanded the study by adding seven
more countries that had similar claims to the "convergence club" in 1870 as those in
Baumol's original set. The new data, when analyzed, didn't align as neatly with the
convergence hypothesis. Statistical analysis by De Long indicated a weaker goodness
of fit, suggesting a less systematic relationship between a country's growth rate and
its initial per capita GDP across the studied countries. He argued that the 1870 data
likely contain large measurement errors, making any measurement of convergence
more inflated than the actual case merits. When adjustments were made to account
for this, the statistical evidence for convergence weakened, highlighting the
challenges in establishing conclusive evidence for unconditional convergence based
on historical data.

A large set of countries over a short time horizon

The second approach is to examine a large set of countries to test for
unconditional convergence. This method aims to minimize statistical
irregularities but requires a shorter time span due to data availability for a
larger group of countries.

In Chapter 2, the Summers–Heston dataset was used to analyze the global

income distribution from 1965 to 1985. The idea of unconditional
convergence seems unlikely based on this analysis. The gap between the
richest and poorest countries doesn't appear to have significantly
narrowed. While the absolute income of the poorest countries has
increased, their relative income disparity with the richest countries has
remained constant, as they've grown at a similar rate.

An objection to this observation is that a comprehensive sample should

cover a broad range of countries, not just the richest and poorest. In
Chapter 2, we acknowledged the diverse experiences of different countries
over time. Grouping countries into clusters and tracking their movements
doesn't show a strong tendency for convergence.

Parente and Prescott (1993) studied 102 countries from 1960 to 1985,
expressing each country's per capita real GDP as a fraction of the U.S. per
capita GDP. The standard deviation of these values increased by 18.5%
over the period, contrary to the expectation of convergence. However,
when looking at regional subgroups, Western European countries showed
a clear decline in relative incomes, while Asian countries exhibited a
significant increase, consistent with data going back to 1900.

Another analysis involves regressing average per capita growth from 1960
to 1985 on per capita GDP in 1960. In a scenario of level convergence, a
negative relationship between these variables would be expected.
However, Barro (1991) found a low correlation of 0.09, suggesting no clear
relationship. Figure 3.10, using Heston–Summers data, illustrates per
capita income in 1960 against the average annual growth rate between
1960 and 1985, showing no discernible pattern in the data.
.

Conditional convergence
Let's consider a significant flaw in the unconditional convergence prediction: the
assumption that all countries share the same level of technical knowledge, savings
rates, population growth rates, and depreciation rates. In reality, countries vary in
these aspects. While the Solow model predicts convergence to steady states, this
weaker hypothesis allows for different steady states, eliminating the need for
countries to converge to each other. This concept is known as conditional
convergence.

In discussing conditional convergence, we maintain the assumption of freely flowing

knowledge across countries, ensuring consistent technological know-how. However,
other parameters like savings rates and population growth rates can differ between
countries. In the Solow model, these parameters only affect the level of per capita
income, not its growth rate, which is solely determined by the rate of technical
progress assumed to be uniform across all countries. This leads to the prediction of
convergence in growth rates, where although long-run per capita incomes vary, the
long-run per capita growth rates are expected to be the same.

Figure 3.11 illustrates convergence in growth rates, depicting various countries with
their own steady-state paths, assumed to be parallel due to the same rate of
technical progress. Suppose a country starts above its steady-state path (AB) at point
C; the Solow model predicts a slower rate of growth over time (curve CD).
Conversely, a country starting below its steady-state path (A′B′) at point E is expected
to exhibit a higher growth rate, with the resulting path EF converging upward to its
steady state.

However, this form of convergence raises a crucial question for practical testing.
Does growth convergence imply that poorer countries tend to grow faster? The
answer is no, as illustrated in Figure 3.11. The country starting at point C is poorer
than the one starting at point E, yet it grows slower. Growth rate convergence
suggests that a country below its steady state grows faster than its steady-state
growth rate, requiring us to identify the positions of these steady states using data.
Unlike unconditional convergence, which assumes all steady states are in the same
place, growth rate convergence needs to be appropriately "conditioned" on the
positions of different steady states.

This concept of controlling for steady-state positions is known as conditional

convergence, where we factor out the effects of varying parameters across countries
before examining the possibility of convergence. The next section will guide us on
how to approach this and control for different parameters.

REEXAMINING THE DATA

To properly apply the concept of conditioning to the data, let's delve into the details
of what the theory is trying to convey. Our goal is to find a relationship between per
capita income and various parameters, such as savings rates and population growth
rates, that can be estimated using the available data.

To find the steady state in the Solow model with technical progress, we solve an
equation involving savings, population growth, and other factors. This process might
seem a bit complex, but I'll simplify it for you. The key takeaway is that we aim to
establish a connection between per capita income and different parameters.

Once we have this relationship, we express it in a form that can be observed directly
from the data. The theoretical framework suggests a specific equation to estimate,
where certain coefficients represent the impact of savings and population growth on
per capita income.

The plan is to examine real-world data on variables like per capita income, savings
rates (s), and population growth rates (n), and regress per capita income on these
parameters. The theory predicts that the coefficients in the regression equation will
be close in value to each other and, more importantly, have opposite signs.

Entering the empirical study with certain expectations:

1. Coefficients on the term ln(s) are expected to be positive, indicating that

savings has a positive effect on per capita income. On the other hand, the
coefficient on the term ln(n + π + δ) is anticipated to be negative, reflecting
the negative effect of population growth on per capita income.
2. The estimated coefficients should have the same approximate magnitude,
around 0.5.

Mankiw, Romer, and Weil [1992] conducted a test using real-world data. While their
results supported some aspects of the Solow model's predictions, such as the
positive impact of savings and the negative effect of population growth on per capita
income, they encountered challenges. The coefficients observed were larger than
anticipated and varied in magnitude.

This observation leads to a significant implication: the differences in per capita

income across countries are larger than predicted by the theory. This discrepancy
suggests tensions between the observed data and the assumptions made by the
Solow model. On one hand, the model implies diminishing returns to capital, while
on the other hand, the large observed differences in per capita incomes suggest
otherwise.
To move forward, we recognize that while dropping the assumption of identical
parameters across countries makes the Solow model more relevant to the data, it
introduces a new challenge. We must now question why savings rates and
population growth rates systematically differ across countries. The assumption that
these differences exist without deeper understanding leaves us with an incomplete
explanation.

Similarly, the assumption of exogenous technical progress is questioned. If technical

progress is a key driver of long-run growth, we need to explore why and how this
progress occurs and whether it seamlessly spreads across countries. This line of
thinking leads us to the newer growth theories

In philosophy, the concept of inequality can lead to endless debates about language
and meaning. However, at its core, economic inequality is the fundamental difference
that allows one person certain material choices while denying those choices to
another. This forms the basis for various scenarios: two individuals earning the same
amount may face different challenges, such as one being physically handicapped.
Even if someone is wealthier, they may lack certain freedoms in their country, like the
right to vote or travel freely. Economic inequality is complex and closely tied to
factors like lifetimes, personal capabilities, and political freedoms.

Despite this complexity, it doesn't mean we can't make meaningful comparisons.

Disparities in personal income and wealth, though narrower compared to broader
issues like freedom and capabilities, still carry significance. This becomes even more
evident when examining economic inequalities within a country, where some broader
issues affect everyone in a similar manner. Therefore, we study income and wealth
inequalities not as a comprehensive measure of all differences but as a crucial
component of overall disparities

Introduction

When there's a lot of income disparity in a society, it's often easy to notice.
For instance, if one person takes the entire cake meant to be shared, it's
unequal; if it's evenly split, it's equal. Determining inequality becomes more
challenging with more individuals and complex divisions of resources. To
address this, we explore the concept of "measuring" inequality, involving
the development or examination of inequality indices. These indices help
rank income or wealth distributions in different situations like countries,
regions, or points in time.
The key question is: What properties should an ideal inequality index
possess? Achieving complete agreement on this is difficult, leading to
various suggested indices with differing results in practical applications.
Balancing weak criteria may yield numerous indices but lacks consensus,
while stricter criteria limit the number of acceptable indices, risking reduced
approval.

This pervasive problem underscores the importance of understanding the

criteria behind a specific measure. Believing in an inequality measure aligns
your intuitive notions of inequality with that measure. For policymakers and
advisors, this alignment can be either beneficial or risky, depending on your
grasp of the underlying measurement criteria.

Population Principle Simplified

The population principle states that duplicating the entire population, along with
their incomes, shouldn't change the level of inequality. In simpler terms, comparing
the income distribution of a group of n people to another group of 2n people with
the same income pattern repeated twice should show no difference in inequality.
This principle emphasizes that population size itself doesn't matter; what matters are
the proportions of the population earning different income levels.

When analyzing income distributions, data sets often present income classes rather
than individual incomes. These classes are typically ranges, like "$100 per month or
less" or "$300–400." The anonymity principle allows us to order people by increasing
income without losing information, and the population principle encourages
normalizing everything to percentages. This way, income classes can be represented
on the horizontal axis, and the percentage of the population in each class on the
vertical axis, simplifying the understanding of inequality trends

Dalton Principle Simplified

The Dalton principle, introduced by Dalton in 1920, is a key criterion for assessing
inequality and plays a fundamental role in constructing measures of inequality.
Suppose we have an income distribution (y1, y2, ..., yn), and consider two incomes, yi
and yj, where yi is less than or equal to yj. If we can transform one income
distribution into another through a series of regressive transfers (shifting income
from the less affluent to the more affluent), the Dalton principle asserts that the
initial distribution is more unequal than the final one.
Now, let's understand how these criteria guide us. To do this, we need to formally
define an inequality measure. This measure is essentially a rule that assigns a level of
inequality to any given income distribution. It takes the incomes (y1, y2, ..., yn) and
produces a value indicating the degree of inequality. A higher value implies greater
inequality.

To meet the anonymity principle, the inequality measure must be insensitive to any
rearrangement of the income distribution among individuals. The population
principle states that cloning the entire population and their incomes should not
impact the inequality measure, allowing us to treat different distributions as if they
have the same population size. The relative income principle is incorporated by
requiring the inequality measure to be unchanged when all incomes are multiplied
by a positive number λ.

Finally, the Dalton transfer principle demands that for any income distribution and
any positive transfer δ, if income is transferred from a less affluent person to a more
affluent person, the resulting distribution should be considered less unequal. This
principle helps capture the impact of changes in income distribution on overall
inequality

Relative Income Principle Simplified

The relative income principle suggests that only the relative differences in incomes
matter for inequality, not the absolute values. If one income distribution can be
obtained from another by scaling everyone's income up or down by the same
percentage, then the inequality should remain the same. In simpler terms, the actual
dollar amounts or currency don't affect the inequality comparison.

With the relative income principle, we can simplify data presentation further.
Expressing both population and incomes as shares of the total allows easy
comparison of income distributions across countries with different average income
levels. In a hypothetical example shown in Figure 6.3, the population is divided into
equal-sized groups, called quintiles, based on their income levels. Income shares for
each quintile are recorded, showing how these shares increase from the poorest to
the richest. The relative income principle tells us that these income shares are
sufficient for measuring inequality, eliminating the need to consider absolute income
values

LORENZ CURVE

Understanding inequality can be facilitated by visualizing it through a Lorenz curve.

This graphical representation is widely used in economic research. Imagine arranging
people in a population based on increasing incomes. The Lorenz curve illustrates
cumulative percentages of the population on the horizontal axis and the percentage
of national income on the vertical axis. Each point on the curve signifies how much
income a certain fraction of the population possesses.

For example, point A might represent the poorest 20% earning only 10% of overall
income. Point B could indicate that the "poorest" 80% enjoy 70% of the national
income. The Lorenz curve connects these points and starts and ends on the 45° line.
If everyone had the same income, the Lorenz curve would coincide with the 45° line.
With increasing inequality, the curve falls below the diagonal, revealing the extent of
inequality in the society.

The slope of the Lorenz curve at any point represents an individual's contribution to
the cumulative share of national income. As we move from left to right, the curve can
never get flatter, emphasizing the constant rise in marginal contribution.

By examining the overall distance between the 45° line and the Lorenz curve, we
intuitively gauge the level of inequality. The greater the distance, the more unequal
the society. No specific formula is needed; a look at the Lorenz curve provides
valuable insights.

The Lorenz criterion helps compare inequality between two distributions. If one
Lorenz curve lies to the right of another at every point, the former is deemed more
unequal. This criterion guides the development of inequality measures, ensuring they
align with this visual assessment of inequality.

Thus an inequality measure I is Lorenz-consistent if, for every pair of income distributions
(y1 , y2 , . . . , yn ) and (z1 , z2 , . . . , zm),

I(y1 , y2 , . . . , yn ) ≥ I(z1 , z2 , . . . , zm)

whenever the Lorenz curve of (y1 , y2 , . . . , yn ) lies everywhere to the right of (z1 ,
z2 , . . . , zm)
Understanding the Connection Between Criteria

The introduction of the Lorenz criterion may seem like an additional layer of
complexity, but there's good news: it is closely linked to the four criteria discussed
earlier. In fact, an inequality measure is consistent with the Lorenz criterion if and
only if it aligns with the anonymity, population, relative income, and Dalton principles
simultaneously.

This observation is beneficial for two reasons. First, it consolidates the earlier four
criteria into the Lorenz criterion, demonstrating their equivalence. Second, it offers a
concise graphical representation of the joint content of these criteria. This graphical
summary simplifies our understanding of ethical criteria for inequality.

Let's delve into why this connection is true. The Lorenz curve inherently incorporates
the principles of anonymity, population, and relative income by focusing on income
and population shares rather than specific magnitudes. Now, let's consider the
Dalton principle. Imagine a regressive transfer from the fortieth to the eightieth
population percentile. According to the Dalton principle, this should increase
inequality.

Examining Figure 6.6 helps visualize this. The thicker curve represents the original
Lorenz curve, and the thinner curve shows the Lorenz curve after the resource
transfer. As expected, the new Lorenz curve dips below and to the right of the old
curve over a certain interval. This dip signifies a temporary increase in inequality due
to the transfer. Eventually, the curves coincide again, indicating that the overall effect
of the transfer has vanished.

In summary, the Lorenz criterion reflects the Dalton principle, as evident in the
bowing of the Lorenz curve. The converse is also true: if two Lorenz curves align
according to the Lorenz criterion, it implies the possibility of constructing
disequalizing transfers between them. This connection streamlines our
understanding and provides a powerful tool for evaluating inequality.

While it seems like we've got everything figured out with our criteria and Lorenz
curves, things get a bit tricky because Lorenz curves can actually cross. This means
that there are situations where neither curve is consistently to the right of the other,
defying the Lorenz criterion and, by extension, our four principles.

In cases like this, it implies that we can't transform one income distribution into the
other solely through a series of Dalton regressive transfers. Instead, there would be
both "progressive" and "regressive" transfers involved in the transformation. An
example helps illustrate this: consider a society with four individuals earning incomes
of 75, 125, 200, and 600. Now, compare it to a second distribution (25, 175, 400, 400).
You can reach the second distribution through a combination of regressive and
progressive transfers, showcasing the inadequacy of the four principles for
comparison.

These complexities lead us to the challenge of weighing the "cost" of regressive

transfers against the "benefit" of progressive transfers, a task that's tough to quantify
in a universally agreeable manner.

Now, when we look at the Lorenz curves in the example, they reflect these intricacies.
For instance, the poorest 25% in the first distribution earn 7.5% of the income, while
in the second, they only get 2.5%. However, as you move to the poorest 75%, their
share goes from 40% in the first to 60% in the second.

Despite these challenges, Lorenz curves offer a clear visual representation of income
distribution in a country. Figure 6.8 even provides Lorenz curves for different
countries, allowing us to visually grasp income inequalities globally and make rough
comparisons between countries
Lorenz curves offer a visual way to grasp the level of inequality in a society. However, there
are two notable challenges with this representation. Firstly, policymakers and researchers
often prefer a numerical summary of inequality for a more concrete and quantifiable
understanding. Secondly, when Lorenz curves intersect, they fail to offer reliable inequality
rankings. Therefore, an inequality measure that assigns a numerical value to every possible
income distribution serves as a comprehensive ranking of income distributions.

The Kuznets ratios, introduced by Simon Kuznets, are indicators used in his influential study
of income distributions in both developed and developing countries. These ratios typically
represent different aspects, such as the share of income held by the poorest 20% or 40% of
the population, or the share owned by the wealthiest 10%. More commonly, they express the
ratio between the income shares of the richest x% and the poorest y%, with x and y taking
values like 10, 20, or 40. These ratios essentially capture segments of the Lorenz curve,
providing a convenient summary in situations where detailed income distribution data may
be lacking

The mean absolute deviation is a measure that considers the entire income
distribution to gauge inequality. It calculates inequality by assessing the
distances of individual incomes from the average income. The process
involves summing up all these income deviations from the mean and then
expressing this sum as a fraction of the total income.

However, the mean absolute deviation has a significant drawback. It tends

to be insensitive to the Dalton principle. For instance, if there are two
individuals with incomes yj and yk, where yj is below the mean income and
yk is above it, a regressive transfer from yj to yk increases inequality
according to the mean absolute deviation. This happens because the
distances of both yj and yk from the mean rise, leading to an unambiguous
increase in the mean absolute deviation.

Yet, the Dalton principle is intended to apply to all regressive transfers, not
exclusively those from incomes below the mean to incomes above it.
Consider any two incomes, yj and yk, both exceeding the mean. If a transfer
occurs from the lower of the two (yj) to the higher one, yk, and is small
enough to keep both incomes above the mean, the sum of the absolute
differences from mean income remains unchanged. In such cases, the mean
absolute deviation shows no variation, thus failing to adhere to the Dalton
principle. Consequently, while it considers the entire income distribution,
the mean absolute deviation lacks compensatory features and is deemed
an inadequate measure of inequality.

The coefficient of variation (C) offers a solution to the insensitivity issue of the mean
absolute deviation by assigning more weight to larger deviations from the mean. In
statistical terms, it utilizes the standard deviation, which involves squaring all
deviations from the mean. Since the square of a number rises more than
proportionately to the number itself, this method effectively gives greater
importance to larger deviations.

The coefficient of variation (C) is derived by dividing the standard deviation by the
mean, ensuring that only relative incomes matter in the calculation. Remarkably, this
measure (C) exhibits satisfactory properties, meeting all four principles and being
Lorenz-consistent. Specifically, it consistently adheres to the Dalton transfer principle.
For instance, in the case of a transfer from j to k, where yj ≤ yk, the square of the
larger number (yk - μ) increases more than the square of the smaller number (yj - μ)
decreases. Consequently, the coefficient of variation (C) consistently records an
increase when such a regressive transfer occurs.