From Mean and Median Resulting Measure of . . . Income to the Most - PowerPoint PPT Presentation

Mean Income and Its . . . Medium Income: A . . . Gauging “Average” . . . From the Idea to an . . . From Mean and Median Resulting Measure of . . . Income to the Most Relation Between the . . . First Example: Case of . . . Adequate Way of Taking Second Example: . . . The New Measure x . . . Inequality Into Account Home Page Title Page Vladik Kreinovich 1 , Hung T. Nguyen 2 , 3 , and Rujira Ouncharoen 4 ◭◭ ◮◮ ◭ ◮ 1 Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA, vladik@utep.edu Page 1 of 17 2 Department of Mathematical Sciences, New Mexico State University Las Cruces, New Mexico 88003, USA, hunguyen@nmsu.edu Go Back 3 Faculty of Economics, 4 Department of Mathematics Full Screen Chiang Mai University, Chiang Mai, Thailand, rujira@chiangmai.ac.th Close Quit

Mean Income and Its . . . Medium Income: A . . . 1. Outline Gauging “Average” . . . • How can we compare the incomes of two different coun- From the Idea to an . . . tries or regions? Resulting Measure of . . . Relation Between the . . . • At first glance, it is sufficient to compare the mean First Example: Case of . . . incomes. Second Example: . . . • However, this is known to be not a very adequate com- The New Measure x . . . parison. Home Page • A more adequate description of economy is the median Title Page income. ◭◭ ◮◮ • However, the median is also not always fully adequate. ◭ ◮ • We use Nash’s bargaining solution to come up with the Page 2 of 17 most adequate measure of “average” income. Go Back • On several examples, we illustrate how this measure Full Screen works. Close Quit

Mean Income and Its . . . Medium Income: A . . . 2. Mean Income and Its Limitations Gauging “Average” . . . • How can we compare the economies of two countries From the Idea to an . . . (or two regions) A and B? Resulting Measure of . . . Relation Between the . . . • At first glance, we can divide the total income by the First Example: Case of . . . number of people, and get mean incomes µ A and µ B . Second Example: . . . • If µ A > µ B , we conclude that A’s economy is better. The New Measure x . . . Home Page • Problem: What if Bill Gates walks into a bar? Title Page • On average, everyone becomes a millionaire. ◭◭ ◮◮ • If a billionaire moves into a poor country, its mean income increases but the country remains poor. ◭ ◮ Page 3 of 17 • So, when comparing different economies, we also need to take into account income inequality. Go Back Full Screen Close Quit

Mean Income and Its . . . Medium Income: A . . . 3. Medium Income: A More Adequate Measure Gauging “Average” . . . • The most widely used alternative to mean is the me- From the Idea to an . . . dian income m A , the level for which: Resulting Measure of . . . Relation Between the . . . – the income of exactly half of the population is above First Example: Case of . . . m A , and Second Example: . . . – the income of the remaining half is below m A . The New Measure x . . . • This is how the Organization for Economic Coopera- Home Page tion and Development (OECD) compares economies. Title Page • Median resolves some of the mean’s problems. ◭◭ ◮◮ • When Bill Gates walks into a bar, median does not ◭ ◮ change much. Page 4 of 17 • In statistical terms: Go Back – the main problem with the mean is that it is not robust, Full Screen – on the other hand, median is robust. Close Quit

Mean Income and Its . . . Medium Income: A . . . 4. Limitations of the Median Gauging “Average” . . . • Example 1: From the Idea to an . . . Resulting Measure of . . . – if the incomes of all the people in the poorer half Relation Between the . . . increase – but do not exceed the previous median, First Example: Case of . . . – the median remains the same. Second Example: . . . • So, median is not adequate measure for describing how The New Measure x . . . well people are lifted out of poverty. Home Page • Example 2: Title Page ◭◭ ◮◮ – if the income of the poorer half drastically decreases, – we should expect the adequate measure of “aver- ◭ ◮ age” income to decrease, Page 5 of 17 – but the median remains unchanged. Go Back Full Screen Close Quit

Mean Income and Its . . . Medium Income: A . . . 5. Gauging “Average” Income Reformulated as a Gauging “Average” . . . Particular Case of Group Decision Making From the Idea to an . . . • Simplest case: all the people in A have the same income Resulting Measure of . . . x , and all the people in B have the same income y . Relation Between the . . . First Example: Case of . . . • If x > y , this clearly country A is better. Second Example: . . . • If x < y , then B’s economy is better. The New Measure x . . . Home Page • In practice, people from A have different incomes x 1 , . . . , x n , and people people from have different incomes y 1 , . . . , y m . Title Page • So, we find x s.t. for A, incomes x 1 , . . . , x n are equiva- ◭◭ ◮◮ lent (in terms of group decision making) to x, . . . , x . ◭ ◮ • Similarly, we find y s.t. incomes y 1 , . . . , y n are equiva- Page 6 of 17 lent to incomes y, . . . , y . Go Back • If x > y , then A’s economy is better, else B’s economy Full Screen is better. Close Quit

Mean Income and Its . . . Medium Income: A . . . 6. From the Idea to an Algorithm Gauging “Average” . . . • Nash showed that the best idea is to to select the al- From the Idea to an . . . n ternative with the largest product of utilities � u i . Resulting Measure of . . . i =1 Relation Between the . . . • The utility is usually proportional to a power of the First Example: Case of . . . money: u i = C i · x a i for some a ≈ 0 . 5. Second Example: . . . n n n The New Measure x . . . x a • Maximizing � � i is equivalent to � x i → max . C i · Home Page i =1 i =1 i =1 • Thus, the equivalent value x comes from the formula Title Page n n ◭◭ ◮◮ � � x = x n . x i = ◭ ◮ i =1 i =1 Page 7 of 17 • The resulting value x is the geometric mean √ x 1 · . . . · x n of the income values. Go Back x = n Full Screen Close Quit

Mean Income and Its . . . Medium Income: A . . . 7. Resulting Measure of “Average” Income Gauging “Average” . . . • Task: compare the economies of regions A and B. From the Idea to an . . . Resulting Measure of . . . • Given: incomes x 1 , . . . , x n in region A. Relation Between the . . . • Given: incomes y 1 , . . . , y m in region B. First Example: Case of . . . • Comparison procedure: Second Example: . . . √ x 1 · . . . · x n The New Measure x . . . – compute the geometric averages x = n Home Page √ y 1 · . . . · y m of the two regions; and y = m Title Page – if x > y , then region A is in better economic shape; ◭◭ ◮◮ – if x < y , then region B is in better economic shape. ◭ ◮ √ x 1 · . . . · x n ) = ln( x 1 ) + . . . + ln( x n ) • ln( x ) = ln( n , so: n Page 8 of 17 �� � x = exp( E [ln( x )]) = exp ln( x ) · f ( x ) dx . Go Back • So, to compare the economies, we need to compare the Full Screen mean values E [ln( x )] of the logarithm of the income x . Close Quit

Mean Income and Its . . . Medium Income: A . . . 8. Relation Between the New Measure and the Gauging “Average” . . . Mean Income: An Observation From the Idea to an . . . • It is well known that the geometric mean is always Resulting Measure of . . . smaller than or equal to the arithmetic mean. Relation Between the . . . First Example: Case of . . . • Geometric mean is equal to the arithmetic mean if and Second Example: . . . only if all the numbers are equal. The New Measure x . . . • Thus, the new measure of “average” income is always Home Page smaller than or equal to the mean income. Title Page • The new measure is equal to the mean income ◭◭ ◮◮ – if and only if all the individual incomes are the ◭ ◮ same, Page 9 of 17 – i.e., if and only if we have perfect equality. Go Back Full Screen Close Quit

Mean Income and Its . . . Medium Income: A . . . 9. First Example: Case of Low Inequality Gauging “Average” . . . • Case: most incomes are close to one another. From the Idea to an . . . Resulting Measure of . . . • Thus, most incomes are close to the mean income µ . Relation Between the . . . • In statistical terms, low inequality means that the stan- First Example: Case of . . . dard deviation σ is small. Second Example: . . . • According to the Taylor series for the logarithm: The New Measure x . . . Home Page ln( x ) = ln( µ +( x − µ )) = ln( µ )+1 µ · ( x − µ ) − 1 2 µ 2 · ( x − µ ) 2 + . . . Title Page ◭◭ ◮◮ • Thus, ignoring higher order terms, ◭ ◮ E [ln( x )] = ln( µ ) − 1 2 µ 2 · σ 2 + . . . Page 10 of 17 Go Back • For x = exp( E [ln( x )]), we similarly get to x = µ − σ 2 2 µ. Full Screen Close Quit