Question

in an imaginary economy, consumers buy only sandwiches and magazines. the fixed basket consists of 20 sandwiches and 30 magazines. in 2006, a sandwich cost $4 and a magazine cost $2. in 2007, a sandwich cost $5. the inflation rate in 2007 was 16 percent. suppose the price index was 105 in 2017, 126 in 2018, and the inflation rate between 2018 and 2019 was lower than it was between 2017 and 2018. this means that multiple choice 1 point the price index in 2019 was lower than 126.0. the price index in 2019 was lower than 147.0. the price index in 2019 was lower than 151.2. the inflation rate between 2018 and 2019 was lower than 1.2 percent.

Explanation:

Step1: Recall inflation - rate formula

The inflation rate formula is $\text{Inflation Rate}=\frac{\text{CPI}_{t}-\text{CPI}_{t - 1}}{\text{CPI}_{t - 1}}\times100\%$, where $\text{CPI}_{t}$ is the consumer - price index in year $t$ and $\text{CPI}_{t - 1}$ is the consumer - price index in the previous year.

Step2: First, find the price of the basket in 2006

In 2006 (base year), the cost of 20 sandwiches at $4$ each and 30 magazines at an unknown price $x$ is $20\times4 + 30x$. Since it's the base - year, CPI = 100.

Step3: Calculate the price of the basket in 2007

The inflation rate in 2007 is 16%. Let the CPI in 2007 be $P_{2007}$. Using the inflation formula with $\text{CPI}_{2006}=100$ and inflation rate = 16%, we have $16=\frac{P_{2007}-100}{100}\times100\%$, so $P_{2007}=116$.

The cost of 20 sandwiches at $5$ each and 30 magazines at price $y$ in 2007 is $20\times5+30y$.

We know that $\text{CPI}=\frac{\text{Cost of basket in current year}}{\text{Cost of basket in base year}}\times100$. Let the cost of the basket in 2006 be $C_{2006}=20\times4 + 30x$ and in 2007 be $C_{2007}=20\times5+30y$. Since $\text{CPI}_{2007} = 116$ and $\text{CPI}_{2006}=100$, we have $\frac{20\times5 + 30y}{20\times4+30x}\times100 = 116$. In the base - year, assume $x$ is the price of a magazine in 2006. Since we are not concerned with the price of sandwiches in the base - year calculation for the magazine price, we can also use the fact that for the magazine part only:

Let the price of a magazine in 2006 be $p_{0}$ and in 2007 be $p_{1}$.

We know that the overall inflation rate affects the price of magazines. If we consider the price of magazines only, using the inflation formula for a single good.

Let the cost of 30 magazines in 2006 be $30p_{0}$ and in 2007 be $30p_{1}$.

The inflation rate formula gives $16=\frac{30p_{1}-30p_{0}}{30p_{0}}\times100\%=\frac{p_{1}-p_{0}}{p_{0}}\times100\%$.

We know that the cost of a magazine in 2007:
Let the cost of a magazine in 2007 be $p$.
We know that the inflation rate formula $\text{Inflation Rate}=\frac{p - p_{base}}{p_{base}}\times100\%$. Given inflation rate = 16% and assume $p_{base}$ is the price of a magazine in 2006.

If we assume the price of a magazine in 2006 is $p_{0}$ and in 2007 is $p_{1}$, then $0.16=\frac{p_{1}-p_{0}}{p_{0}}$, so $p_{1}=(1 + 0.16)p_{0}$.

We are given that in 2006, assume the price of a magazine is $p_{0}$ and in 2007, using the inflation rate of 16% for the magazine price.

If we assume the price of a magazine in 2006 is $p$, then the price of a magazine in 2007, $P$ is given by $P=(1 + 0.16)p$.

We know that the cost of a magazine in 2007:
Let the cost of a magazine in 2006 be $x$. The cost of a magazine in 2007, $y$ is related to $x$ by $y=(1 + 0.16)x$.

We can also use the CPI formula for the basket. The cost of the basket in 2006 is $C_{2006}=20\times4+30x$ and in 2007 is $C_{2007}=20\times5+30y$. Since $\frac{C_{2007}}{C_{2006}}\times100 = 116$.

Simplifying, we get $\frac{100 + 30y}{80+30x}=1.16$.

Since in the base - year, we can assume some normalization, and using the inflation rate for a single good (magazine) formula:

If the inflation rate is 16%, and let the price of a magazine in 2006 be $p$, then the price of a magazine in 2007, $P=p(1 + 0.16)$.

We know that the answer can be found by using the inflation rate formula for a single good.
If we assume the price of a magazine in 2006 is $p$, and the inflation rate is 16%, then the price of a magazine in 2007, $P$ is:
$P=\frac{116}{100}\times p$.

Let's assume the price of a magazine in 2006 is $p$. Then in 2007, with an inflation rate of 16%, the price of a magazine is $p_{2007}=(1 + 0.16)p$.

If we consider the fact that for a 16% inflation rate, the new price of a magazine is $P=1.16p$.

We know that the cost of a magazine in 2007 is calculated as follows:
Let the price of a magazine in 2006 be $x$. Then the price of a magazine in 2007, $y = 1.16x$.

If we assume the price of a magazine in 2006 is $2$, then the price of a magazine in 2007 is $2\times1.16 = 2.32$.

Answer:

$2.32$