Chapter 14, Problem 14.58ACB

Insured Social Security workers. Workers insured under the Social Security program are categorized as fully and permanently insured, fully but not permanently insured, or insured in the event of disability. The number of workers (in millions) in each insured category from 2000 to 2015 are provided in the accompanying table.

a. Compute a simple composite index for the number of workers in the three insured categories using 2000 as the base period.

b. Is the index, part a, a price index or a quantity index?

c. Refer to part a. Interpret the index value for 2015.

To determine

To compute: The simple composite index for the number of workers in the three insured categories using 2000 as the base period.

Answer to Problem 14.58ACB

The simple composite index for the number of workers in the three insured categories using 2000 as the base period is,

Year	Fully Permanent	Fully Not Permanent	Event Disability	Index
2000	140.9	44.9	139.5	100.0
2001	142.9	45.2	141.7	101.4
2002	144.9	45.3	143.5	102.6
2003	147	45	144.9	103.6
2004	149	44.8	146.2	104.5
2005	151.1	44.7	147.7	105.6
2006	153.3	45.1	150.1	107.1
2007	155.4	45.6	152.3	108.6
2008	157.4	46	154.5	110.0
2009	159.2	44.8	150.6	109.0
2010	161.1	44.6	151.7	109.9
2011	163.1	44.3	152.6	110.7
2012	163.3	44	149.2	109.6
2013	165.3	43.9	149.9	110.4
2014	167.3	43.8	150.6	111.2
2015	169.4	43.8	151.2	112.0

Explanation of Solution

Given info:

The table shows that the number of workers in different categories from 2000 to 2015.

Calculation:

The total number of workers in fully and permanently insured, fully but not permanently insured or insured in the event of disability of the years 2000-2015 is obtained as follows:

The total number of workers in fully and permanently insured, fully but not permanently insured or insured in the event of disability of the 2000 is obtained by adding the number of workers in fully and permanently insured, fully but not permanently insured or insured in the event of disability of the year 2000.

$\begin{array}{c}\text{Sum of 2000}=140.9+44.9+139.5\\ =325.3\end{array}$

Similarly the totals of the remaining years 2001-2015 is as follows:

Year	Fully Permanent	Fully Not Permanent	Event Disability	Sum
2000	140.9	44.9	139.5	325.3
2001	142.9	45.2	141.7	329.8
2002	144.9	45.3	143.5	333.7
2003	147	45	144.9	336.9
2004	149	44.8	146.2	340.0
2005	151.1	44.7	147.7	343.5
2006	153.3	45.1	150.1	348.5
2007	155.4	45.6	152.3	353.3
2008	157.4	46	154.5	357.9
2009	159.2	44.8	150.6	354.6
2010	161.1	44.6	151.7	357.4
2011	163.1	44.3	152.6	360.0
2012	163.3	44	149.2	356.5
2013	165.3	43.9	149.9	359.1
2014	167.3	43.8	150.6	361.7
2015	169.4	43.8	151.2	364.4

The formula to find the simple composite index is,

$I_t=\frac{\text{Sum value at time }t}{\text{Time series value at base period}}\times 100$

For year 2000:

The simple composite index for the number of workers in the three insured categories for year 2000 using 2000 as the base period is as follows:

Substitute 325.3 for “Sum value at time t” and 325.3 for “Time series value at base period”

$\begin{array}{c}I=\frac{325.3}{325.3}\times 100\\ =100.00\end{array}$

Similarly, the simple composite index for the remaining years using 2000 as the base year is as follows:

Year	Fully Permanent	Fully Not Permanent	Event Disability	Sum	Simple Composite Index-2000
2000	140.9	44.9	139.5	325.3	100.0
2001	142.9	45.2	141.7	329.8	101.4
2002	144.9	45.3	143.5	333.7	102.6
2003	147	45	144.9	336.9	103.6
2004	149	44.8	146.2	340.0	104.5
2005	151.1	44.7	147.7	343.5	105.6
2006	153.3	45.1	150.1	348.5	107.1
2007	155.4	45.6	152.3	353.3	108.6
2008	157.4	46	154.5	357.9	110.0
2009	159.2	44.8	150.6	354.6	109.0
2010	161.1	44.6	151.7	357.4	109.9
2011	163.1	44.3	152.6	360.0	110.7
2012	163.3	44	149.2	356.5	109.6
2013	165.3	43.9	149.9	359.1	110.4
2014	167.3	43.8	150.6	361.7	111.2
2015	169.4	43.8	151.2	364.4	112.0

To determine

To identify: Whether the index in part (a) is price index or quantity index.

Answer to Problem 14.58ACB

The index in part (a) is quantity index.

Explanation of Solution

Justification:

From the given information it can be observed that the index in part (a) represents the quantities. Thus, the index in part (a) is quantity index.

To determine

To interpret: The index value for 2015.

Answer to Problem 14.58ACB

The index value for 2015 interprets that the total number of insured workers in 2015 is increased by 12.0% when the base year is 2000.

Explanation of Solution

Justification:

Here, it is observed that the total number of insured workers in 2015 is increased by 12.0% $\left(=120.0-100\right)$ when the base year is 2000.