Appendix: The Lorenz curve

Introduction

The Lorenz curve is a graphical device used to demonstrate the equity of distribution of a given variable such as income, asset ownership or wealth. For example, one might be interested in the equity of cattle ownership since this is often taken as an indicator of the distribution of wealth, particularly for pastoral and agropastoral societies. The distribution of cattle ownership is, however, often extremely difficult to determine so that the cattle holding per household (or per holder) is often used as the proxy measure of wealth in the derivation of a Lorenz curve. Holding implies the right and responsibility to manage on a day-to-day basis but not necessarily to dispose of (e.g. by slaughter, sale or gift).

In the following example, a Lorenz curve for cattle holdings to households holding cattle is therefore constructed. The principles outlined in the derivation of the curve can be applied to any data set in which the equity of distribution for a given variable is being calculated.

Derivation of the Lorenz curve

In the derivation of the example Lorenz curve, the following procedure has been adopted:

· All individual units (households) are ranked from the lowest to the highest according to the number of cattle held (Column 1, Table 2.A1) and the number of households in each cattle-holding category is given (Column 2).

· From this data, the percentage of households falling into each cattle-holder category is derived (Column 3).

· The cumulative percentage of households in each cattle-holder category is then estimated (Column 4).

· By multiplying the number of cattle in each category by the number of households holding those cattle (Column 1 x Column 2), we then obtain the total number of cattle held within each category (Column 5).

· From this latter figure, the percentage of total cattle held (Column 6) and the cumulative per cent of cattle held in each category (Column 7) are obtained.

· The cumulative percentage of cattle held in each category (vertical axis) is then plotted against the cumulative percentage of households for each category (horizontal axis) to derive the Lorenz curve (Figure 2A.1).

· This plotted curve is then compared with the line of perfect equity (drawn at 45° from the origin of the graph) to provide an indication of the equity of distribution of cattle holdings within the area concerned.

Table 2A.1. Data required for derivation of the Lorenz curve of cattle holdings to households.

(1) Cattle-holding category (no. of cattle held)	(2) No. house-holds holding cattle in each category	(3) % households holding cattle in each category	(4) Cumulative % of households in each category	(5) Total no. of cattle in each category	(6) % total cattle within each category	(7) Cumulative % of cattle within each category
0	30	23.07	23.07	0	0.00	0.00
1	5	3.85	26.92	5	0.73	0.73
2	5	3.85	30.77	10	1.46	2.19
3	5	3.85	34.62	15	2.20	4.39
4	7	5.38	40.00	28	4.10	8.49
5	8	6.15	46.15	40	5.86	14.35
6	10	7.69	53.84	60	8.78	23.13
7	12	9.23	63.07	84	12.30	35.43
8	15	11.54	74.61	120	17.57	53.00
9	15	11.54	86.15	135	19.77	72.77
10	12	9.23	95.38	120	17.57	90.34
11	6	4.62	100.00	66	9.66	100.00
	130	100.00		683	100.00

Exercise 2.A: Drawing and interpreting a Lorenz curve (estimated time required: 20 minutes).

Question 1. Using the data in Table 2.3, draw, on the same piece of paper, Lorenz curves for the distribution of cattle per cattle-owning households and per cattle-owning person.

Question 2. Is the distribution of cattle per cattle-owning people more or less inequitable than the distribution per cattle-owning household? Use the Lorenz curves drawn in response to Question 1 and the note to Figure 2A.1 to provide you with the answer.

Note to Figure 2A.1: The 45° line drawn from the origin of the graph represents the line of perfect equity where the percentage of cattle held corresponds exactly to the percentage of households in the area, where each household has the same number of cattle, i.e. where 10% of the households hold 10% of the cattle etc. The more bowed the Lorenz curve from this line (indicated by the hatched area in Figure 2A.1), the more inequitable is the distribution of cattle holdings. The calculation of equivalent Lorenz curve data for two different areas, regions or countries can provide a useful basis for equity comparisons.

Figure 2A.1. Lorenz curve of cattle holdings to households holding cattle.

Exercise 2.1: Group exercise: Identification of policy issues: Central case study (estimated time required: 1.5 hours).

Question 1. Read the central case study carefully and list the various criteria/problems confronting the livestock sector in Alphabeta. Where possible, cite evidence from the report for the criteria/problems you have identified.

Question 2. What evidence can you obtain from the report to demonstrate that the government's objectives are similar to those given in the lecture notes?

Question 3. On the basis of your previous answers, what policy issues (criteria/problems) would you now isolate for more detailed analysis? Why?

Question 4. For the particular issues you have isolated as most important, what broad policy strategies/instruments could be appropriate? How would you make a start in an evaluation/analysis of these alternative strategies? What strategies/instruments has the government of Alphabeta adopted? Are they effective? If not, why not?

Exercise 2.2: Identification of policy issues - Country specific study (estimated time required: 1 hour).

Question 1. What are your country's broad objectives for the livestock sector? Whose objectives are they?

Question 2. In what document, or equivalent, are these objectives set out? How do you know these are your country's objectives? Are they declared or undeclared objectives? Which are more important?

Question 3. In your country, are there specific goals or targets set for the livestock sector? How? Where?

Question 4. Which of the criteria/problems listed in the lecture notes are relevant to your country? Are there problems which you consider urgent in your country? Could you assign priorities to your country's problems in the livestock sector?

Question 5. Which policies/instruments does your country pursue in order to solve the most urgent problems listed in your answer to Question 4? Are these instruments appropriate? Identify some other instruments which might be used instead to solve the same problems.

Question 6. What is your professional responsibility? How does it relate to overall policy objectives? What particular policy issues would you prefer to see discussed in this manual and why?