§4.6 Applications to Economics.

§4.6 Applications to Economics. The student will learn about: consumers’ surplus, producers’ surplus, and the Gini index.

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.

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.

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.

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 B = d (A) Market Price Where d (x) is the price demand function.

Consumers’ Surplus - Example 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 = A is given as 150 and the market price B = d (A) = d (150) A is given as 150 and the market price B = d (A) = d (150) = 300 – 150/2 = 225 The customers have paid $5,625 less than they were willing to pay. A “savings”. $ 5,625

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.

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.

Lorenz Curve The Lorenz curve may be compared with two extreme cases of income distribution. 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. 9

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. 10

Lorenz Curve In reality the Lorenz curve is somewhere between those two extremes.

Gini Index of Income Distribution 12

A measure of 0 indicates absolute equality. A measure of .5 indicates absolute inequality. 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.

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. 14

Index of Income Concentration If y = f (x) is the Lorenz curve, then Index of income concentration = Income Distribution - Application Double the area bounded by y = x and the Lorenz curve is called the index of income concentration. 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.

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.

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 = 0.3939 Future: Index of income concentration = 0.20

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.

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.

Income Distribution – Facts of Interest Gini Coefficients in the World

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.

Summary. We used our previous knowledge of the area between two curves to: To calculate consumers’ surplus, and To find the “Gini” index..

ASSIGNMENT §4.6 on my website.

$ Consumers’ surplus Graphically Price Market price B = d (A) d (x) x A Quantity 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.

§4.6 Applications to Economics.