Background Information Scientists studying the economics of a country, community
ID: 2831623 • Letter: B
Question
Background Information
Scientists studying the economics of a country, community, or group of people are often interested in the distribution of wealth, where wealth is understood to mean the income or the personal assets of individual households. A powerful tool for illustrating how wealth is distributed across a society is the Lorenz curve.
Directions
A typical Lorenz curve is given by y = L9x), where 0 ? x ? 1 and 0 ? y ? 1. The variable x represents the fraction of all households in the society and y = L(x) represents the fraction of the total wealth that is owned by the fraction x of the society. For example, for the Lorenz curve shown in Figure 1, we see that L(0.5) = 0.2, which means that 0.5 (50%) of the society owns 0.2 (20%) of the wealth.
1. Figure 1 shows that L(0.8) = 0.6. Interpret the point (0.8, 0.6) on the Lorenz curve.
2. A Lorenz curve y = L(x) is always accompanied by the line y = x, called the line of perfect equality. Explain why this line is given this name.
3. Explain why any Lorenz curve must have the following properties:
a. L(0) = 0 and L(1) = 1
b. L is an increasing function on [0, 1] and
c. The graph of L must lie on or below the line of perfect equality. In most cases, the Lorenz curve is concave up on the interval [0, 1].
4. A family of functions that nicely describes theoretical Lorenz curves is L(x) = xp, where p ? 1. Verify that these functions have the properties listed in Step 3.
Explanation / Answer
1. The point (0.8,0.6) shows that 80% of all households control 60% of all wealth.
2. The line y=x is called the line of perfect equality because for every point on this line, the values of the x and y co-ordinate are always equal.
3.a) It must have L(0) = 0 because 0% of the wealth will be controlled by 0% of the families. In other words, if there are no people in a nation, there will be no wealth.
It must have L(1) = 1 because the complete sum of all the nation's wealth is being controlled by all the families combined.
b) L must be increasing in [0,1] because as the no of families controlling the quantity of wealth naturally increases if the wealth increases.
c) It must always lie beneath the line of equality because the line of equality represents an ideal case which can never actually happen in the real world. It is concave upward because its derivative (rate of change) keeps increasing.
4. y=xp. where p>1. as is shown in the figure.
Putting in various values of p i.e 2,3 etc we get curves like x2, x3 etc. As can be seen from plotting these curves in the intervals x = [0,1] and y = [0,1], all the properties of Lorenz curve are satisfied.
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