§4.6 Applications to Economics.
The student will learn about:
consumers’ surplus,
producers’ surplus, and
the Gini index.
1
Introduction
The applications we are going to discuss today
are all applications that use the area between
two curves, the topic we covered during the
last class period.
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Consumers’ Surplus
Our first application is consumers’ surplus.
Imagine you were working hard and wanted
your favorite beverage. You were willing to
pay $3.00 for that beverage. However, the cost
of the beverage is only $2.50. You have
“saved” 50¢. If you sum that 50¢ over all of
the consumers that purchase the beverage you
have the consumers’ surplus.
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Mathematical Definition of Consumers’ Surplus
The demand curve gives the price that consumers
are willing to pay, and the market price is what they
do pay, so the amount by which the demand curve
is above the market price measures the benefit or
“surplus” to consumers.
This total benefit (the shaded
area in the diagram on the
right) is called the consumers’
surplus.
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Consumers’ Surplus
Customers’ Surplus – Given a
demand function and a demand
level x = A, the market price B is the
demand function evaluated at x = A,
so that B = d (A). The consumers’
surplus is the area between the
demand curve and the market
price. Or
Consumers' Surplus 
B = d (A)
Market Price
 d(x)  d(A) dx
A
0
Where d (x) is the price demand function.
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Consumers’ Surplus - Example
Consumers' Surplus 
 d(x)  d(A) dx
A
0
For the demand function d (x) = 300 – x/2, find the
consumers’ surplus at a demand level of x = 150.
A is given as 150 and the market price
B = d (A) = d (150) = 300 – 150/2 = 225
Consumers ' Surplus  
150
0


x
  300 - 2  - 225  dx 



x

   75 -  dx  $ 5,625 The customers have paid
0
2

$5,625 less than they were willing to pay. A “savings”.
150
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Gini Index of Income Distribution
In any society, some people make more money
than others.
To measure the “gap” between the rich and the
poor, economists calculate the proportion of the
total income that is earned by the lowest 20% of
the population, and then the proportion that is
earned by the lowest 40% of the population, and
so on.
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Gini Index of Income Distribution
This information (for the United States in the year 2006)
is given in the table below (with percentages written as
decimals), and is graphed on the right.
For example, the lowest 20% of the population earns only 3%
of the total income, the lowest 40% earns only 12% of the
total income, etc. The curve is known as the Lorenz curve.
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Lorenz Curve
The Lorenz curve may be compared with two extreme cases
of income distribution.
1. Absolute equality of income
means that everyone earns
exactly the same income, and
so the lowest 10% of the
population earns exactly
10% of the total income, the
lowest 20% earns exactly
20% of the income, and so on.
This gives the Lorenz curve y = x shown on the right.
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Lorenz Curve
2. Absolute inequality of income
means that nobody earns any
income except one person,
(Bill Gates) who earns
all the income.
This gives the Lorenz curve
shown on the right.
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Lorenz Curve
In reality the Lorenz curve is somewhere
between those two extremes.
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Gini Index of Income
Distribution
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Income Distribution - Application
The area bounded by y = x and the Lorenz curve is the
area under the line y = x and above the Lorenz curve
from x = 0 to x = 1 . The smaller that area the more
even the income is distributed.
A measure of 0 indicates
absolute equality.
A measure of .5 indicates absolute inequality.
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Gini Index
To measure how the actual distribution differs from
absolute equality, we calculate the area between the
actual distribution (The Lorenz Curve) and the line of
absolute equality y = x.
Since this area can be at most ½ (the area of the entire
lower triangle), economists multiply the area by 2 to
get a number between 0 and 1.
This measure is called the Gini index.
Note that a higher Gini index means greater inequality.
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Income Distribution - Application
Double the area bounded by y = x and the Lorenz curve
is called the index of income concentration.
Index of Income Concentration
If y = f (x) is the Lorenz curve, then
 2 x  f (x) dx
1
Index of income concentration =
0
This measure is called the Index of Income
Concentration or the Gini Index.
A measure of 0 indicates absolute equality. A measure of
1 indicates absolute inequality.
15
Index of Income Concentration
To calculate the Index of Income of Concentration we
evaluate the following integral where y = f (x) is the
Lorenz curve for the income distribution.

1
0
2 [ x  f (x) ]dx
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Example
A country is planning changes in tax structure in
order to provide a more equitable distribution of
income. The two Lorenz curves are:
f (x) = x 2.3 currently and g (x) = 0.4x + 0.6x 2 proposed.
Will the proposed changes work?
Currently: Index of income concentration =

1
0
2 [ x  x 2.3 ] dx  0.3939
Future: Index of income concentration =

1
0
2 [ x  (0.4 x  0.6 x 2 )] dx  0.20
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Income Distribution – Facts of
Interest
The index of income concentration is sometimes called
the Gini Coefficient. Countries with a Gini Coefficient
between 0.5 and 0.7 are regarded as having unequal
income distribution while countries having coefficients
between 0.2 and 0.35 are considered to have relatively
equitable. In addition, coefficients are calculated for
rural areas verses urban areas. And coefficients are
plotted over time to look for trends in economic
distribution.
18
Income Distribution – Facts of
Interest
During the past thirty-seven years the United States’s
Gini Coefficient has gone from 0.39 to 0.46.
Unless the United States breaks this trend, the
American middle class will be a thing of the past actually within the lifetime of most Americans living
today.
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Income Distribution – Facts of
Interest
Gini Coefficients in the World
Hungary
Denmark
0.244
0.247
Portugal
China
0.385
0.447
Germany
Canada
Australia
0.283
0.331
0.352
United States
Panama
Namibia
0.463
0.564
0.707
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Income Distribution
Facts of Interest
I also found evidence that the Gini Coefficient is used
in analysis of items in a warehouse and in measuring
health inequalities by the Pan American Health
Organization. In the second instance one of the uses
was to compare the cumulative proportion of live
births to the cumulative proportion of infant deaths.
21
Summary.
We used our previous knowledge of the area
between two curves to:
To calculate consumers’ surplus, and
To find the “Gini” index..
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ASSIGNMENT
§4.6 on my website.
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Consumers’ Surplus
The demand function d (x) and the market price A must
be given in the problem. Then B = d (A), and you can
plug the information into the equation.
Consumers' Surplus 
 d(x)  d(A) dx
A
0
$
Price
Graphically
Consumers’
surplus
Market price
B = d (A)
d (x)
A
x
Quantity
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