Daphne Rutnam
Candidate number: 001482-0018
Inequality in my local borough
Inequality is a prominent aspect of the world today. There are stark contrasts between developed
and undeveloped countries as well as within countries where the rich can be many times wealthier
than the poor.
In my borough, Islington, the divide between the rich and the poor is particularly visible through the
contrast between the deteriorating council estates and the luxurious new apartments that have
sprung up. Despite a reputation for its upmarket boutiques and coffee shops, Islington is the 14th
most deprived borough in England and has the second highest rate of child poverty.1 45% of children
and young people in the borough live in poverty.2
There are various ways of measuring inequality in a region, but a common statistical method is to
draw a Lorenz curve and to calculate the corresponding Gini coefficient in order to give each region a
particular value relating to the level of inequality in that region or country.
I decided I would try to create a Lorenz curve for Islington and calculate the Gini coefficient. Learning
about inequality is important as it can inform us about a place and the people who live there. In
addition, there is growing evidence that inequality leads to other social problems.3 For example,
Islington has the second highest rate of crime of all London boroughs, with 10.45 crimes per 100,000
people.4 In this year alone, 4 people have died as a result of knife crime in Islington.5 By doing this
research, I hope to gain a greater insight into my local area.
Figure 1: A
picture of
Islington
1
Williams, R., Islington Council Fairness Commision Wealth Gap, 2011,
http://www.theguardian.com/society/2011/jun/08/islington-council-fairness-commission-wealth-gap, last
viewed: 1/11/15
2
Islington Council, Two Islingtons: Understanding the Problem, 2012,
http://www.islington.gov.uk/publicrecords/library/Democracy/Information/Factsheets/2011-2012/(2012-0303)-What_is_the_picture.pdf, last viewed: 1/11/15
3
New Economics Foundation, Distant Neighbours, 2013,
http://b.3cdn.net/nefoundation/5756b988b34063f6c9_ltm6is8u9.pdf, last viewed: 10/11/15
4
Danby, P., Islington has second highest crime rate of all London boroughs, 2015,
http://islingtonnow.co.uk/2015/03/25/islington-has-second-highest-crime-rate-of-all-london-boroughs/, last
viewed: 10/11/15
5
Dodd, V., Stabbed teenager is 18th to be murdered in London this year, http://www.theguardian.com/uknews/2015/nov/24/teenager-stabbed-holloway-18th-to-die-in-london-this-year
1
Daphne Rutnam
Candidate number: 001482-0018
The Lorenz Curve
The Lorenz curve is a graphical representation of the cumulative proportion of wealth owned by a
cumulative proportion of the population. I decided to measure wealth as household income, a
common measure of wealth in Lorenz curves.
The red line 𝑦 = 𝑥 shows a perfectly equal distribution of income as everyone has the same income.
The blue curve and green curve show 2 examples of Lorenz curves for 2 different countries. Country
2’s Lorenz curve sits below Country 1’s Lorenz curve.
While the blue line shows that the highest-earning 20% owns 60% of the wealth, the green line
shows that the highest-earning 20% owns 80% of the wealth.
There is therefore a greater degree of inequality in Country 2 as a larger proportion of total income
belongs to a smaller proportion of the population than in Country 1.
Total income
My first task was to find out the total annual income in Islington. The mean household income in
2008 was £57,000.6 The number of households was around 80,5007.
57000 × 80500 = 4.5885 × 109
Therefore total income is around £4.6 billion.
6
Butterworth, M., Average salaries rise to more than £31,000, survey shows, The Telegraph, 2008
http://www.telegraph.co.uk/finance/personalfinance/3531599/Averagesalaries-rise-to-more-than-31000survey-shows.html, last viewed: 10/11/15
7
Islington Council, Islington Census Summary, 2013, http://www.vai.org.uk/wpcontent/uploads/2013/01/2012-Census-Islington-Summary.pdf, last viewed: 10/11/15
2
Daphne Rutnam
Candidate number: 001482-0018
Income distribution
Secondly, I needed to find out how income was distributed amongst the households of Islington. I
used a pie chart issued by Islington council to create my initial set of income brackets.
Figure 2: Household income,
2008
Income / £000s
Households / %
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 60
60 ≤ x
However, in order to create a more accurate Lorenz curve, I needed to create smaller income
brackets.
15
32
36
17
Median income
I discovered that the median income for Islington households was £32,000.8 The median income
describes the income at which there are equal numbers of households earning below it and above it.
So, 50% of households have an income of less than £32,000. I added this data to my set of income
brackets.
Income / £000s
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 32
32 ≤ x < 60
60 ≤ x
Households / %
15
32
3
33
17
8
Department for Work and Pensions, London Borough of Islington Profile, 2010, www.dwp.gov.uk/docs/cpaislington.xls, last viewed: 10/11/15
3
Daphne Rutnam
Candidate number: 001482-0018
Highest earners
I also discovered that 11% of households have incomes greater than £75,000.9 I added this statistic
to the set of income brackets.
Income / £000s
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 32
32 ≤ x < 60
60 ≤ x < 75
75 ≤ x
Households / %
15
32
3
33
6
11
Income for each of these income brackets
The next step was to find the total income for each of these income brackets. Firstly, I calculated the
number of households in each of these income brackets, using the fact that there were a total of
80,500 households in the borough and using the proportions of households which fell into each of
these income brackets. My results were as follows:
Income / £000s
Households / %
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 32
32 ≤ x < 60
60 ≤ x < 75
75 ≤ x
15
32
3
33
6
11
Households (80500 x
Proportion of households)
12075
25760
2415
26565
4830
8855
Secondly, I had to calculate the average income for each of these income brackets. I did this in three
ways, each of which gave a different picture of inequality in Islington.
9
Islington Council, Islington Tenancy Strategy, 2015,
http://www.islington.gov.uk/publicrecords/library/Housing/Businessplanning/Strategies/2012-2013/(2013-0117)-Islington-Tenancy-Strategy-2012-15.pdf, last viewed: 10/11/15
4
Daphne Rutnam
Candidate number: 001482-0018
Method 1: assuming average income to be midpoint for each bracket
Assuming that incomes were roughly uniformly distributed over each income bracket, I firstly
calculated the midpoint, m, for income for all of the brackets by performing the simple calculation:
𝑚=
𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 + 𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡
2
I could not use the highest bracket as it has no upper limit. I then worked out the total income for
each of the income brackets, using m and the number of households. For the highest earners, I
calculated the total income from subtracting the total of other incomes from the total borough
income, which I had calculated earlier from the mean borough income.
After doing this, I had to calculate the cumulative shares of wealth owned by cumulative proportions
of the population. I calculated cumulative income by income bracket and then I calculated the
proportion of cumulative income that represented.
Income /
£000s
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 32
32 ≤ x < 60
60 ≤ x < 75
75 ≤ x
Household
s/%
15
32
3
33
6
11
Cumulative
households / %
15
47
50
83
89
100
Household
s
12075
25760
2415
26565
4830
8855
m/
£000s
7.5
22.5
31.0
46.0
67.5
Total income /
£000s
90562.5
579600.0
74865.0
1221990.0
326025.0
2295457.5
Cumulative
income / £000s
90562.5
670162.5
745027.5
1967017.5
2293042.5
4588500.0
Plotting the points
Along the x-axis of a Lorenz curve is the cumulative number of households, and along the y-axis is
the cumulative proportion of income. These two sets of values are highlighted in blue in my data
table. I then plotted the points:
Method 2: assuming average income to be highest possible value for each bracket
The second method I used involved assuming the average income of each income bracket to be the
highest possible value. For example, in the 32,000≤x<60,000 bracket, I assumed that the average
income was £60,000. This assumption was probably not accurate, but I hoped this method would
help to paint a picture of the minimum possible inequality in Islington in order to help ascertain a
more probable Lorenz curve and Gini coefficient.
5
Cumulative
income / %
1.97
14.61
16.24
42.87
49.97
100.00
Daphne Rutnam
Candidate number: 001482-0018
I calculated the total income for each bracket using the upper limit and the number of households in
that bracket. I then calculated the total income of the highest-earning households by subtracting the
total incomes of all lower-earning households from the total borough income, like in method 1.
Results were as follows:
Income /
£000s
Household
s/%
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 32
32 ≤ x < 60
60 ≤ x < 75
75 ≤ x
15
32
3
33
6
11
Cumulative
households / %
15
47
50
83
89
100
Household
s
12075
25760
2415
26565
4830
8855
Upper
limit
/£000s
7.5
22.5
31.0
46.0
67.5
Total income /
£000s
90562.5
579600.0
74865.0
1221990.0
326025.0
2295457.5
Cumulative
income / £000s
90562.5
670162.5
745027.5
1967017.5
2293042.5
4588500.0
Cumulative
income / %
1.97
14.61
16.24
42.87
49.97
100.00
Plotting the points
Method 3: assuming average income to be lowest possible value for each bracket
Like method 2, assuming the average income for each bracket to be the highest possible income is
rather unrealistic, but I wanted to create a more detailed picture of possible levels of inequality in
Islington so I used this method anyway. I expected it to show a very high level of inequality.
Here are the results:
Income /
£000s
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 32
32 ≤ x < 60
60 ≤ x < 75
75 ≤ x
Household
s/%
15
32
3
33
6
11
Cumulative
households / %
15
47
50
83
89
100
Household
s
12075
25760
2415
26565
4830
8855
6
Lower
limit /
£000s
7.5
22.5
31.0
46.0
67.5
Total income /
£000s
90562.5
579600.0
74865.0
1221990.0
326025.0
2295457.5
Cumulative
income / £000s
90562.5
670162.5
745027.5
1967017.5
2293042.5
4588500.0
Cumulative
income / %
1.97
14.61
16.24
42.87
49.97
100.00
Daphne Rutnam
Candidate number: 001482-0018
Plotting the points
Evaluating the different methods
Method 3 clearly shows the highest degree of inequality as the highest-earning 10% of households
own around 60% of total income. Method 2 shows the lowest degree of inequality as the highestearning 10% of households own only around 30% of income, half of what they would earn according
to method 3. Both method 2 and 3 are relatively unlikely, as we would not expect everyone within
each income bracket to be earning the maximum possible amount for that bracket, nor would we
expect that everyone is earning the least possible amount for that bracket. I will therefore use the
values given by Method 1 to continue my calculations to find the Gini coefficient.
7
Daphne Rutnam
Candidate number: 001482-0018
Drawing the Lorenz curve
The next step was to fit a function to the points to create a Lorenz curve for Islington. The graph
would have to go through the point (100,100).
Quadratic function
1
I firstly drew the function 𝑦 = 100 𝑥 2 . This would go through (100,100). This is seen in grey.
The grey curve sits above many of the points and is therefore not a very good fit and would indicate
a much lower level of inequality than data in Islington would show.
Cubic function
I then decided to try out a cubic equation. I used the function 𝑦 =
1
𝑥3
10000
to model a potential
curve as it would go through the point (100,100). This cubic function is seen in red.
8
Daphne Rutnam
Candidate number: 001482-0018
The cubic graph gives a better indication of how the shape of the Lorenz curve of Islington might
look. However, it misses out several points and suggests a more uniform distribution of income
amongst the wealthiest households than might be expected in Islington.
Quadratic function
1
I then decided to draw a quadratic function, 𝑦 = 1000000 𝑥 4 which is seen in green.
Again, this is not a particularly good fit as the quadratic curve lies below the bottom points and
above the top points.
A more accurate equation of the Lorenz Curve for Islington
I then decided to use Autograph to generate a line of best-fit which allowed me to have a more
accurate Lorenz Curve, as seen in blue:
𝑦 = 6.35 × 10−6 𝑥 4 − 0.000965𝑥 3 + 0.0470𝑥 2 − 0.412𝑥 + 0.145
Although it is not perfect, the curve is better than the other functions and so I decided to use the
equation generated to calculate the Gini coefficient.
9
Daphne Rutnam
Candidate number: 001482-0018
Calculating the Gini coefficient
The equation for the Gini coefficient is:
𝐺=
𝐴
𝐴+𝐵
Where A is the area between the line of perfect equality and the Lorenz curve and B is the area
under the Lorenz curve.
100
𝐴+𝐵 =∫
𝑥 𝑑𝑥
0
𝐴 + 𝐵 = 5000
𝐴 = 5000 − 𝐵
𝐺=
5000 − 𝐵
5000
Because B is the area under the Lorenz curve, we can integrate:
100
𝐵=∫
𝑓(𝑥) 𝑑𝑥
𝑛
Where 𝑓(𝑥)is the equation of the Lorenz Curve. 𝑛 is the point where the graph crosses the y-axis. In
the case of Islington,
𝑓(𝑥) = 6.35 × 10−6 𝑥 4 − 0.000965𝑥 3 + 0.0470𝑥 2 − 0.412𝑥 + 0.145
6.35 × 10−6 𝑛4 − 0.000965𝑛3 + 0.0470𝑛2 − 0.412𝑛 + 0.145 = 0
𝑛 = 10.636
100
𝐵=∫
6.35 × 10−6 𝑥 4 − 0.000965𝑥 3 + 0.0470𝑥 2 − 0.412𝑥 + 0.145 𝑑𝑥
10.636
= [1.27 × 10
100
965
47 3 103 2
29
4
𝑥 −
𝑥 +
𝑥 −
𝑥 +
𝑥]
4000000
3000
500
2000 10.636
−6 5
= 2201.99
𝐺=
5000 − 2201.99
5000
𝐺 = 0.560
10
Daphne Rutnam
Candidate number: 001482-0018
Conclusion
I started out with the aim of finding out more about the levels of inequality in my local borough. I
discovered that the Gini coefficient of my borough, Islington, was 0.56. This is a relatively large
number and shows that Islington has high inequality, with many households earning very little and a
small number earning a large amount. The high Gini coefficient is comparable to countries such as
Zambia, the Central African Republic and Honduras, some of the most unequal countries in the
world.
The Lorenz curve shows that in Islington, 50% of total income is owned by less than 10% of the total
number of households. By contrast, the poorest 50% own only 16% of total income.
This is evidence of the high proportion of very rich and very poor people living in Islington, like in the
rest of London, which houses a disproportionately large share of the country’s poorest and richest
people.10 However, the Gini coefficient is far higher in Islington than in London as a whole which is
around 0.4411.
There are several limitations to my investigation. I had to assume that mean income for each income
bracket was the midpoint, which is unlikely to be the case. I also neglected the fact that some
households may be in debt and so the actual Lorenz curve may start below the y-axis.
I have been left with a real desire to find out more about the effects of inequality on my area and to
do similar research on other boroughs in London, to find comparable data on levels of inequality.
10
Adomaitis, K., The World’s Largest Cities Are The Most Unequal, 2013,
http://blog.euromonitor.com/2013/03/the-worlds-largest-cities-are-the-most-unequal.html, last viewed:
10/11/15
11
Ibid
11
Daphne Rutnam
Candidate number: 001482-0018
Bibliography/references
Figure 1: New Economics Foundation, Distant Neighbours, 2013,
http://b.3cdn.net/nefoundation/5756b988b34063f6c9_ltm6is8u9.pdf, last viewed: 10/11/15
Figure 2: Islington Council, Two Islingtons: Understanding the Problem, 2012,
http://www.islington.gov.uk/publicrecords/library/Democracy/Information/Factsheets/20112012/(2012-03-03)-What_is_the_picture.pdf, last viewed: 1/11/15
1. Adomaitis, K., The World’s Largest Cities Are The Most Unequal, 2013,
http://blog.euromonitor.com/2013/03/the-worlds-largest-cities-are-the-most-unequal.html,
last viewed: 10/11/15
2. Butterworth, M., Average salaries rise to more than £31,000, survey shows, The Telegraph,
2008 http://www.telegraph.co.uk/finance/personalfinance/3531599/Averagesalaries-rise-tomore-than-31000-survey-shows.html, last viewed: 10/11/15
3. Danby, P., Islington has second highest crime rate of all London boroughs, 2015,
http://islingtonnow.co.uk/2015/03/25/islington-has-second-highest-crime-rate-of-all-londonboroughs/, last viewed: 10/11/15
4. Department for Work and Pensions, London Borough of Islington Profile, 2010,
www.dwp.gov.uk/docs/cpa-islington.xls, last viewed: 10/11/15
5. Dodd, V., Stabbed teenager is 18th to be murdered in London this year,
http://www.theguardian.com/uk-news/2015/nov/24/teenager-stabbed-holloway-18th-todie-in-london-this-year
6. Islington Council, Islington Census Summary, http://www.vai.org.uk/wpcontent/uploads/2013/01/2012-Census-Islington-Summary.pdf, last viewed: 10/11/15
7. Islington Council, Islington Tenancy Strategy, 2015,
http://www.islington.gov.uk/publicrecords/library/Housing/Businessplanning/Strategies/201
2-2013/(2013-01-17)-Islington-Tenancy-Strategy-2012-15.pdf, last viewed: 10/11/15
8. Islington Council, Two Islingtons: Understanding the Problem, 2012,
http://www.islington.gov.uk/publicrecords/library/Democracy/Information/Factsheets/20112012/(2012-03-03)-What_is_the_picture.pdf, last viewed: 1/11/15
9. New Economics Foundation, Distant Neighbours, 2013,
http://b.3cdn.net/nefoundation/5756b988b34063f6c9_ltm6is8u9.pdf, last viewed: 10/11/15
10. Williams, R., Islington Council Fairness Commision Wealth Gap, 2011,
http://www.theguardian.com/society/2011/jun/08/islington-council-fairness-commissionwealth-gap, last viewed: 1/11/15
12