The share of aggregate income held by middle-income households in 1970 was 62%,

Question

The share of aggregate income held by middle-income households in 1970 was 62%, whereas that held by upper-income households was 29%. The corresponding figures in 2014 were 43% and 49%, respectively. The models describing the fall and the rise in the share of the aggregate incomes of these two groups are approximately linear over the period under consideration. (a) Find the mathematical models describing the percent share of aggregate income held by each group from 1970 through 2014. (Let t denote the time, in years, with t = 0 corresponding to the beginning of 1970.) middle-income households y = upper-income households y = (b) Find the time when the aggregate income held by upper-income households first exceeded that held by middle-income households. (Give the calendar year in which the change occurred.)

Explanation / Answer

Let y1 and y2 denote the share of aggregate held by middle-income households and upper-income households respectively.

Then we have two equations to model our problem

y1 = a1t + b1 and y2 = a2t + b2 since it is given that the relation with time is linear for the percent share of aggregate income held by each group from 1970 through 2014 and here a1 , a2 , b1 , b2 are constant coeffecients .

Now since for the calender year 1970 t =0 so putting that we get

y1 = b1 and y2 = b2 where for the year 1970 y1 = 62 and y2 = 29 so  b1 = 62 and b2 = 29

And we also have the data for the year 2014 for which t will be 2014-1970 = 44 and for the year 2014

y1 = 43 and y2 = 49

So we two equation,

43 = 44* a1  + 62 which is 44* a1 = -19 therefore a1 = 0.432

And the second equation 49 = 44 *  a2 + 29 which is 44 *  a2 = 20 therefore a2 = 0.455  

Therefore our model which describes the  percent share of aggregate income held by each group from 1970 through 2014 are the two following equations

y1 = -0.432t + 62 and y2 = 0.455t + 29

Ans b) The time when the aggregate income held by upper-income households first exceeded that held by middle-income households is basically to calculate

y1 = -0.432t + 62 < y2 = 0.455t + 29 or simply  -0.432t + 62 < 0.455t + 29

which can be written as (0.455 + 0.432)*t > 33

or t > 33 / 0.887 or t > 37.204 years

So according to calender year it is 1970 + 37.204 which is 2007.304 or basically we can say it occured in year 2007

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