Transcript The Math and the Magic of Financial Derivatives

On The Mathematics of
Income Inequality
Klaus Volpert
Villanova University
Sep. 19, 2013
Published in American Mathematical Monthly, Dec 2012
Stunning income differences
In 2010, hedge fund manager
John Paulson took home
$5 Billion.
Stunning income differences
In 2010, hedge fund manager
John Paulson took home
$5 Billion.
That would be the combined income of
50,000 professors -
Stunning income differences
In 2010, hedge fund manager
John Paulson took home
$5 Billion.
That would be the combined income of
50,000 professors if we made an average of $100,000.
That’s more than all the math
professors in the US combined
On the other hand, there is no doubt that one
could find 50,000 of the world's poor whose
combined wealth is less than each one of ours.
On the other hand, there is no doubt that one
could find 50,000 of the world's poor whose
combined wealth is less than each one of ours.
Does it have to be that way?
Total Equality is not possible
Total Equality is not possible
Even if we could distribute all wealth equally, inequality would
return in an instant.
Total Equality is not possible
Even if we could distribute all wealth equally, inequality would
return in an instant.
For one man would take his money to the bank, and one man
would take it to the bar.
Herr Procher (my 10th grade English teacher)
Inequality might be inevitable,
it might be stunning,
But it is not static!
Inequality varies a great deal from country to country.
Even within the same country it can change dramatically over time!
Q: What do you think: over the last 100 years, has
inequality in the US increased or decreased?
Q: What do you think: over the last 100 years, has
inequality in the US increased or decreased?
20
18
Share of total income (in percent) of top 1% Earners
in the US, 1913-2012
16
14
12
10
8
6
4
0
1913
1915
1917
1919
1921
1923
1925
1927
1929
1931
1933
1935
1937
1939
1941
1943
1945
1947
1949
1951
1953
1955
1957
1959
1961
1963
1965
1967
1969
1971
1973
1975
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
2009
2011
2
Data by Thomas Piketty and Emanuel Saez
A better measure than the top 1%-index:
The Lorenz Curve and the Gini-Index
L(x)= share of total
income earned by all
households combined
that are poorer than the
xth percentile.
0.1
e.g., L(.4)=.1 means
that the poorest 40% of
the population have a
share of 10% of the
total income.
The curve has two anchors: (0,0) and (1,1)
The Lorenz Curve and the Gini-Index
If income was perfectly
evenly distributed, then
L(x)=x, and the graph
would be a straight line
Perfect equality
The Lorenz Curve and the Gini-Index
Perfect
inequality
If one household
received all the income,
and no one else
received anything,
the curve would look like
this
The Lorenz Curve and the Gini-Index
Realistic curves are
somewhere in
between.
The closer to the
diagonal the more
equally distributed the
income.
The Lorenz Curve and the Gini-Index
The Gini-Index is the ratio
of the area between the
line of equality and L(x),
and the entire area under
the line of equality
(which is ½).
So, 1
G  2 x  L( x )dx
0
Perfect Equality → G=0
Perfect Inequality → G=1
Example: If L( x)  x2 , then
1
2
3
1
x
x
1
G  2  x  x dx  2(  ) 
2
3 0 3
0
2
Example: If L( x)  x2 , then
1
2
3
1
x
x
1
G  2  x  x dx  2(  ) 
2
3 0 3
0
2
So the Gini-index is a
summary measure,
giving weight to the
poor as to the wealthy
Example: If L( x)  x2 , then
1
2
3
1
x
x
1
G  2  x  x dx  2(  ) 
2
3 0 3
0
2
So the Gini-index is a
summary measure,
giving weight to the
poor and to the wealthy
This allows us to
make comparisons
between countries,
as well as look at
trends over time.
The Gini-Index for various Countries:
(according to data from the World Bank and CIA)
The Gini-Index for various Countries:
(according to data from the World Bank and CIA)
Country
Gini-Index
Sweden
.230
Germany
.270
Egypt
.344
Russia
.420
China
.480
Brazil
.519
South Africa
.650
The Gini-Index for various Countries:
(according to data from the World Bank and CIA)
Country
Gini-Index
Sweden
.230
Germany
.270
Egypt
.344
Russia
.420
China
.480
Brazil
.519
South Africa
.650
Discussion: there are several reasons why one must take
these numbers with a grain of salt. . .
(what are the basic economic units?
What constitutes income?
What about adjustments for cost of living? Etc.)
Still. Overall the comparative impression is correct.
What about the US?
Here are Some Raw Data from the Census
Bureau:
US-Income Data 1967:
Published by the Census Bureau
Quintile
How can we calculate the Giniindex from these Quintiles??
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
Connect the Dots??
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
Connect the Dots??
US-Income Data 1967:
Published by the Census Bureau
Quintile
This would clearly underestimate the
area and the Gini-Index.
How can we do better??
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
A First Model for the Lorenz Curve:
Matched with real income data,
however, no power function fits
the whole population very well:
Best fit:
L( x)  x
2.287
L( x)  x
p
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
L( x)  cx
p
However!
If we only try to match the
bottom 60%, then it does fit well!
We’ll come back to that. . . .
Best fit:
L( x)  .849 x
1.90
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
A Second Model for the Lorenz Curve: A Quintic Polynomial:
L( x)  ax  bx2  cx3  dx4  ex5
A quintic polynomial fits much
better:
US-Income Data 1967:
Published by the Census Bureau
Quintile
L( x)  .13x  .23x2  3.9 x3  5.8x4  2.99 x5
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
A Second Model for the Lorenz Curve: A Quintic Polynomial:
L( x)  ax  bx2  cx3  dx4  ex5
A quintic polynomial fits much
better:
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
L( x)  .13x  .23x2  3.9 x3  5.8x4  2.99 x5
2 Problems: (a) We calculate G=.391, while the official number is G=.397
(b) The coefficients tell us nothing.
Q: Can we come up with a function that has some economic meaning?
The rich don’t feel so rich. . .
The self-similarity phenomenon:
In 2007: (according to data from Saez and Piketty)
•The top 10.0% received 48% of total US income
•The top 1.0% received 24%
•The top .1% received 12%
•The top .01% received 6%
The inequality repeats among the rich:
no matter how rich you are, there are always some who are far richer,
in a very peculiar and predictable pattern.
Q: what function for L(x) would describe such a pattern?
L( x)  1  (1  x) , 0  q  1
q
Here is a family of such
functions.
q=1 → total equality
q=0 → total inequality
This family of functions
is known as Pareto
distributions
The Principle of Self-Similarity `to the right`
These functions have this remarkable property:
The Gini-index among the top x percent is the
same as in the whole population, for any x.
=
=
This can be called a (one-sided) `scale-invariance’,
also known as the `Pareto Principle’.
This function works great for the top 5%:
but, alas, even this model does not fit very well for
the whole population:
So what to do? We have a model that fits very well at the top, and,
remember, we have a model that fits very well at the bottom. . . Let’s
p
take another look at
L( x)  x
L( x)  x
p
exhibits the opposite phenomenon:
Self-similarity to the left:
The inequality repeats among the poor
The Gini-Index among the whole population is the same as the
Gini-Index restricted to just the poorest 50%, say.
Or the poorest 20%. Or the poorest 2%. Etc.
So one model works well at the top, one at the bottom, so for the whole population we
try a hybrid model, that asymptotically equals the two separate models at either end:
The Hybrid Model:
L( x)  x  (1  (1  x) )
p
q
Note: asymptotically we have
as x  0, L( x)  qx
p 1
as x  1, L( x)  (1  (1  x)q )
Empirically, the model fits
amazingly well:
L( x)  x  (1  (1  x) )
p
1967
L( x)  x.7438 (1  (1  x).6823 )
Sum of Squares of Residuals: 0.00001989
q
With this model, G=.395.
Official value:
G=.397
L( x)  x  (1  (1  x) )
p
2008
2008
L( x)  x.7987 (1  (1  x).5608 )
Sum of Squares of Residuals: 0.00000396
q
With this model, G=.468
Official value:
G=.466
Quintic model: G=.457
The Trend 1967-2010
We matched this model to
every year 1967-2010.
Each time the predicted
Gini-index, based on just 4
data points each year,
is within 0.25% of the
official Gini-Index computed
by the Census bureau from
a full set of raw data.
Notice that the latest
(2012) Gini-Index for
the US is
approximately .477
L( x)  x  (1  (1  x) )
p
q
Notice:
p and q are
economically
meaningful:
2008
2008
𝑝
𝐺0 =
𝑝+2
and
L( x)  x
.7987
(1  (1  x)
.5608
)
1−𝑞
𝐺1 =
1+𝑞
are the limits of
the Gini-Index at
the low and high
end respectively
1967-2010: The Trends in 𝐺0 and 𝐺1
Inequality among the poor
has slightly increased
Inequality among the rich has
increased significantly
Concluding Thought
It is easy to forget among all the models and data, that there are real
people affected by poverty and inequality.
Concluding Thought
It is easy to forget among all the models and data, that there are real
people affected by poverty and inequality.
When we think of inequality, do we think of him or her?
Is our thinking, our respect, our care, as skewed as the Lorenz curve?
Concluding Thought
So I would like to finish with a quote from a girl at a
school in South Africa that is under the care of the
Augustinian Mission, which in turn is connected to
Villanova University
Andile
Student, St. Leo’s School, Durban, South Africa
“I am somebody. I may be poor, but I am somebody.
I maybe make mistakes, but I am somebody.”
Andile
Student, St. Leo’s School, Durban, South Africa