Explanation Given: y = F ( r ) is the fraction of total income that goes to the bottom r t h part of the population, where 0 ≤ r ≤ 1 . Concept Used: The national average income is A = T N N is the total number of household r is the total income earned by the entire population. Calculation: Total income that goes to the bottom r t h part = F ( r ) T Total population in this part of population= = N r Average income will be Average = F ( To determine (b) To show: The average income of households belonging to an interval [ r , r + Δ r ] = ( F ( r + Δ r ) − F ( r ) Δ r ) A To determine (c) To show: Pass to the limit as △ r → 0 in [ r , r + Δ r ] = ( F ( r + Δ r ) − F ( r ) Δ r ) A to derive the interpretation: A household in the 100r t h percentile has income F ′ ( r ) A . To determine (d) To show: For the Lorenz curves L 1 and L 2 what percentage of household have above-average income.

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Differential Equations

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Question

Chapter 3.4, Problem 49E

To determine

(a)

To show- The average income among households in the bottom r^t^h part is equal to (F(r)r)A.

To determine

(b)

To show:

The average income of households belonging to an interval [r,r+Δr]=(F( r+Δr)−F(r)Δr)A

To determine

(c)

To show:

Pass to the limit as △r→0 in [r,r+Δr]=(F( r+Δr)−F(r)Δr)A to derive the interpretation: A household in the 100r^t^h percentile has income F′(r)A.

To determine

(d)

To show:

For the Lorenz curves L₁ and L₂ what percentage of household have above-average income.

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Chapter 3.4, Problem 48E

Chapter 3.4, Problem 50E

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Chapter 3 Solutions

Calculus - Standalone book

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Prob. 20E Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Prob. 26E Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Prob. 37E Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Prob. 45E Ch. 3.1 - Prob. 46E Ch. 3.1 - Prob. 47E Ch. 3.1 - Prob. 48E Ch. 3.1 - Prob. 49E Ch. 3.1 - Prob. 50E Ch. 3.1 - Prob. 51E Ch. 3.1 - Prob. 52E Ch. 3.1 - Prob. 53E Ch. 3.1 - Prob. 54E Ch. 3.1 - Prob. 55E Ch. 3.1 - 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(a) To show- The average income among households in the bottom r t h part is equal to ( F ( r ) r ) A .

Solution Summary: The author concludes that the average income among households in the bottom r t h part is equal to y=F(r).

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(a)

To show- The average income among households in the bottom r^t^h part is equal to (F(r)r)A.

(b)

To show:

The average income of households belonging to an interval [r,r+Δr]=(F( r+Δr)−F(r)Δr)A

(c)

To show:

Pass to the limit as △r→0 in [r,r+Δr]=(F( r+Δr)−F(r)Δr)A to derive the interpretation: A household in the 100r^t^h percentile has income F′(r)A.

(d)

To show:

For the Lorenz curves L₁ and L₂ what percentage of household have above-average income.

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Chapter 3 Solutions

(a) To show- The average income among households in the bottom r t h part is equal to ( F ( r ) r ) A .

Solution Summary: The author concludes that the average income among households in the bottom r t h part is equal to y=F(r).

Concept explainers

Videos

(a) To show- The average income among households in the bottom rth part is equal to (F(r)r)A.

(b) To show: The average income of households belonging to an interval [r,r+Δr]=(F( r+Δr)−F(r)Δr)A

(c) To show: Pass to the limit as △r→0 in [r,r+Δr]=(F( r+Δr)−F(r)Δr)A to derive the interpretation: A household in the 100rth percentile has income F′(r)A.

(d) To show: For the Lorenz curves L1 and L2 what percentage of household have above-average income.

Want to see the full answer?

Chapter 3 Solutions

(a)

To show- The average income among households in the bottom r^t^h part is equal to (F(r)r)A.

(b)

To show:

The average income of households belonging to an interval [r,r+Δr]=(F( r+Δr)−F(r)Δr)A

(c)

To show:

Pass to the limit as △r→0 in [r,r+Δr]=(F( r+Δr)−F(r)Δr)A to derive the interpretation: A household in the 100r^t^h percentile has income F′(r)A.

(d)

To show:

For the Lorenz curves L₁ and L₂ what percentage of household have above-average income.