Chapter 6, Problem 14P

(a)

To determine

Determine the change in cost of living using the Laspeyres price index.

Explanation of Solution

Since the price of sunscreen is $1, the quantity consumed is 5 ounces, the price of lemonade is $1.30, and the quantity consumed is 20 gallons, the cost of bundle in June can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\times 5\right)+\left(\text{\$1}\text{.30}\times \text{20}\right)\\ =\$5+\$26\\ =\$31\end{array}$

Thus, the total cost in June at June’s price is $31.

Since $31 is the cost of the base bundle in the base month, the Laspeyres price index in June can be calculate as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{31}{31}\\ =1\end{array}$

Thus, the price index for June is 1.

Now, the cost of June’s bundle at July’s price can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\text{.10}\times 5\right)+\left(\text{\$1}\text{.70}\times \text{20}\right)\\ =\$5.5+\$34\\ =\$39.50\end{array}$

Thus, the total cost of June’s bundle at July’s price is $39.50.

Therefore, the cost difference from June to July is $\$8.50\left(\$39.50-\$31\right)$. 

The cost difference in percentage term can be calculated as follows:

$\begin{array}{c}{\text{Cost}}_{\text{Diff}}=\frac{8.50}{31}\times 100\\ =27.41\%\end{array}$

Thus, there is 27.41% cost difference from June to July.

Since $31 is the cost of the base bundle in the base month, the Laspeyres price index in July can be calculate as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{39}{31}\\ =1.274\end{array}$

Thus, the price index for July is 1.274. 

The cost of June’s bundle at August’s price can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\text{.40}\times 5\right)+\left(\text{\$1}\text{.80}\times \text{20}\right)\\ =\$7+\$36\\ =\$43\end{array}$

Thus, the total cost of June’s bundle in August is $43.

Therefore, the cost difference from July to August is $\$3.50\left(\$43-\$39.50\right)$. 

The cost difference in percentage term can be calculated as follows:

$\begin{array}{c}{\text{Cost}}_{\text{Diff}}=\frac{3.50}{39.5}\times 100\\ =8.86\%\end{array}$

Thus, there is 8.86% cost difference from July to August.

The cost difference from June to August is $\$12\left(\$43-\$31\right)$. 

The cost difference in percentage term can be calculate as follows:

$\begin{array}{c}{\text{Cost}}_{\text{Diff}}=\frac{12}{31}\times 100\\ =38.7\%\end{array}$

Thus, there is 37.7% cost difference from June to August.

Since $31 is the cost of the base bundle in the base month, the Laspeyres price index in August can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{43}{31}\\ =1.387\end{array}$

Thus, the price index for August is 1.387. 

(b)

To determine

Determine the change in the cost of living using the Laspeyres price index.

Explanation of Solution

July (as base month):

Total cost in July at July’s price can be calculated as follows using the similar method explained in Part (a):

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\text{.10}\times 7\right)+\left(\text{\$1}\text{.70}\times 18\right)\\ =\$7.7+\$30.6\\ =\$38.30\end{array}$

Thus, the total cost in July at July’s price is $38.3.

Since July is the base month, $38.3 is the cost of the base bundle. The Laspeyres price index in July can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{38.3}{38.3}\\ =1\end{array}$

Thus, the price index for July is 1.

Now, the cost of July’s bundle in June can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\times 7\right)+\left(\text{\$1}\text{.30}\times 18\right)\\ =\$7+\$23.4\\ =\$30.4\end{array}$

Thus, the total cost of July’s bundle in June is $30.4.

Therefore, the cost difference from June to July is $\$7.90\left(\$38.30-\$30.40\right)$. 

The cost difference in percentage term can be calculated as follows:

$\begin{array}{c}{\text{Cost}}_{\text{Diff}}=\frac{7.90}{30.4}\times 100\\ =25.98\%\end{array}$

Thus, there is 25.98% cost difference from June to July.

Since $38.30 is the cost of the base bundle in the base month, the Laspeyres price index in July can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{30.4}{38.3}\\ =0.7937\end{array}$

Thus, the price index for June is 0.7937.  

The cost of July’s bundle in August can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\text{.40}\times 7\right)+\left(\text{\$1}\text{.80}\times 18\right)\\ =\$9.8+\$32.4\\ =\$42.2\end{array}$

Thus, the total cost of July’s bundle in August is $42.2

Therefore, the cost difference from July to August is $\$3.9\left(\$42.2-\$38.3\right)$. 

The cost difference in percentage term can be calculated as follows:

$\begin{array}{c}{\text{Cost}}_{\text{Diff}}=\frac{3.90}{38.3}\times 100\\ =10.18\%\end{array}$

Thus, there is 10.18% cost difference from July to August.

Since $38.30 is the cost of the base bundle in the base month, the Laspeyres price index in August can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{42.2}{38.3}\\ =1.1018\end{array}$

Thus, the price index for August is 1.1018. 

August (as base month):

Total cost in August at August’s price can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\text{.40}\times 3\right)+\left(\text{\$1}\text{.80}\times 23\right)\\ =\$4.2+\$41.4\\ =\$45.6\end{array}$

Thus, the total cost in August at August’s price is $45.6.

Since August is the base month, $45.6 is the cost of the base bundle. The Laspeyres price index in July can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{45.6}{45.6}\\ =1\end{array}$

Thus, the price index for July is 1.

Now, the cost of August’s bundle in June can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\times 3\right)+\left(\text{\$1}\text{.30}\times \text{23}\right)\\ =\$3+\$29.9\\ =\$32.9\end{array}$

Thus, the total cost of August’s bundle in June is $32.9.

Since $45.6 is the cost of the base bundle in the base month, the Laspeyres price index in July can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{32.9}{45.6}\\ =0.7214\end{array}$

Thus, the price index for June is 0.7214. 

The cost of August’s bundle in July can be calculated as follows:

$\begin{array}{c}\text{Cost}=\left({\text{Price}}_{\text{s}}\times {\text{Quantity}}_{\text{s}}\right)+\left({\text{Price}}_{\text{L}}\times {\text{Quantity}}_{\text{l}}\right)\\ =\left(\text{\$1}\text{.10}\times 3\right)+\left(\text{\$1}\text{.70}\times 23\right)\\ =\$3.3+\$39.1\\ =\$42.4\end{array}$

Thus, the total cost of August bundle in July is $42.4

Since $45.6 is the cost of the base bundle in the base month, the Laspeyres price index in August can be calculated as follows:

$\begin{array}{c}\text{Price index}=\frac{{\text{Cost}}_{\text{june}}}{{\text{Cost}}_{\text{Base}}}\\ =\frac{42.4}{45.6}\\ =.9298\end{array}$

Thus, the price index for July is .9298.

(c)

To determine

Determine whether the answers in Part (a) and Part (b) has any similarities.

Explanation of Solution

Part (a) and Part (b) explain the value of Laspeyres indexes with June as the base year, July as the base year, and August as the base year. As shown above, Laspeyres indexes show different percent changes in the cost of living over time. However, the direction of the changes is the same for each index.