Measuring the Cost of Living

.title[
# Measuring the Cost of Living
]
.subtitle[
## Macroeconomics for Students of Accounting, Finance and Digital Applications
]
.author[
### Lyuben Ivanov, PhD; Georgy Ganev, PhD
]
.institute[
### Sofia University St Kliment Ohridski
]
.date[
### Mar 21, 2024
]

---

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.font130[
.pull-right[
_By a continuing process of inflation, government can confiscate, secretly and unobserved, an important part of the wealth of their citizens._
<hr style="background-color: black; margin: 0em 0em 0em 0em;">
<span style="float: right; font-variant: small-caps; ">John Maynard Keynes &nbsp;</span>
]
]

---

<hr>

---

# Importance of Measuring Cost of Living

.font130[
Contemporary economies are based on division of labor,
specialization and ensuing trade, and trade is based on monetary
exchange, i.e. on payment with money at market prices.
Therefore most of economic agents’ incomes are expressed as some
sum of money.
]

.font130[
So far it is precisely the type of economic organization based on
market monetary exchange which has led to the highest level of
satisfaction of human wants with available scarce resources in history.
]

.font130[
The separation of the moment of a sale of a good or resource from
the moment of a purchase of another good or resources made
possible by the opportunity to hold on to some sum of money means
that the problem with money preserving their value (ability to
purchase real goods or resources) through time becomes very
important.
]

---

# Importance of Measuring Cost of Living

<br>

.font130[
One part of any change in a price (and thus in the value of
money) is due to the constant fluctuation of relative prices
between goods and resources. However, these fluctuations by themselves do not affect the
overall value of money and its purchasing power very much.
]

.font130[
The important factor for the overall purchasing power of
money is the change in the general level of all prices, and this is
why it is precisely this overall price level and its changes which
is the important macroeconomic indicator.
]

.font130[
From the point of view of economic agents, especially in their
role as consumers, the question about the overall price level
boils down to the question of cost of supporting their way of
life.
]

---

# The Consumer Price Index (CPI)
<br>

.font130[
The inflation rate you are likely to hear in the news is calculated from the consumer price index (CPI), which
reflects the average level of prices of the goods and services bought by consumers.
]

.font130[
<table class="none">
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b>consumer price index (CPI)</b></td>
    <td> </td>
    <td> </td>
    <td>a measure of the overall cost of the goods and services bought by a typical consumer.</td>
  </tr>
</table>  
]

.font130[
Knowing the values of CPI for different years allows economists to compare the value of economic variables measured in current prices in different time periods. That is, to adjust nominal values for different years for the effects of changing prices.
]

---

<hr>

---

# CPI and Inflation Rate Calculation
<br>

.font130[
To calculate the CPI and the inflation rate, government statisticians follow 5 basic steps:
]

---

# 1. Fix the Basket
<br>

.font130[
Statisticians observe consumer behavior to determine which prices are most important to the typical
consumer in order to fix the basket.
]

.font130[
If the typical consumer buys more hot dogs than hamburgers,
then the price of hot dogs is more important than the price of hamburgers
and, therefore, should be given greater weight in measuring the cost of
living.
]

.font130[
The statisticians sets these weights by surveying consumers to find the basket
of goods and services bought by the typical consumer.]

.font130[
In the example that follows, we assume that the typical consumer buys a basket of 6 hot dogs and 4 hamburgers.
]

---

# 2. Find the Prices
<br>

.font130[
Here are the prices of each of the goods and services in our hypothetical basket of 6 hot dogs and 4 hamburgers for three different years:
]

<br>

.font130[
<table class = "none">
 <thead>
  <tr>
   <th style="text-align:center;"> Year </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Price of hot dogs ($) </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Price of hamburgers ($) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 2021 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2022 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2023 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 2 </td>
  </tr>
</tbody>
</table>
]

---

# 3. Compute the Basket's Cost
<br>

.font130[
The table below shows the calculation of the cost of the basket of 6 hot dogs and 4 hamburgers
for each of the three years:
]

.font100[
<table class="none">
 <thead>
  <tr>
   <th style="text-align:center;"> Year </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Price of hot dogs ($) </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Quantity of hot dogs </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Price of hamburgers ($) </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Quantity of hamburgers </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Cost of basket ($) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 2021 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 6 </td>
   <td style="text-align:center;"> + </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 34 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2022 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 6 </td>
   <td style="text-align:center;"> + </td>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 36 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2023 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 6 </td>
   <td style="text-align:center;"> + </td>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 38 </td>
  </tr>
</tbody>
</table>
]

.font130[
**NB**: Please note that the prices are changing through the years, but the quantities remain fixed.
]

---

# 4. Choose Base Year and Compute Index

<br>

.font130[
The table below shows the calculation of the consumer price index for each of the three years, using 2021 as the base year:
]

<table class="none">
 <thead>
  <tr>
   <th style="text-align:center;"> Year </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Cost of basket (current year) </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Cost of basket (base year) </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> CPI </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 2021 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 34 </td>
   <td style="text-align:center;"> / </td>
   <td style="text-align:center;"> 34 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 100 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2022 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 36 </td>
   <td style="text-align:center;"> / </td>
   <td style="text-align:center;"> 34 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 106 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2023 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 38 </td>
   <td style="text-align:center;"> / </td>
   <td style="text-align:center;"> 34 </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 112 </td>
  </tr>
</tbody>
</table>

.font130[
**NB**: Please note that the CPI for the base year is always 100 because the current year and the base year are 
the same, implying that the cost of the basket in the current year is the same as the cost of the basket in the base year.
]

---

# 5. Compute the Inflation Rate

.font130[
<table class="none">
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b>inflation rate</b></td>
    <td> </td>
    <td> </td>
    <td>the percentage change in the price index from the preceding period.</td>
  </tr>
</table>  
]

<br>
<table class="none">
<thead>
  <tr>
   <th style="text-align:center;"> Year </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> CPI </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> CPI (previous year) </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;">   </th>
   <th style="text-align:center;"> Inflation rate </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 2021 </td>
   <td style="text-align:center;"> ( </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> / </td>
   <td style="text-align:center;"> NA </td>
   <td style="text-align:center;"> - </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> ) </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> NA </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2022 </td>
   <td style="text-align:center;"> ( </td>
   <td style="text-align:center;"> 106 </td>
   <td style="text-align:center;"> / </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> - </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> ) </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 6% </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2023 </td>
   <td style="text-align:center;"> ( </td>
   <td style="text-align:center;"> 112 </td>
   <td style="text-align:center;"> / </td>
   <td style="text-align:center;"> 106 </td>
   <td style="text-align:center;"> - </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> ) </td>
   <td style="text-align:center;"> x </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> = </td>
   <td style="text-align:center;"> 6% </td>
  </tr>
</tbody>
</table>

--
<br>
<div style="text-align:right"> **Mathematical note:** &nbsp; &nbsp; &nbsp; </div>
$$
\% \Delta X = \frac{\Delta X}{X}\times 100 = \frac{X_2 - X_1}{X_1}\times 100 = (\frac{X_2}{X_1} - 1)\times 100
$$

---

# General Formula for CPI Calculation

<br>
.font130[

`$$\begin{aligned}
\sf CPI_t  & = \sf \frac{\sum_{i=1}^n P^i_{t} Q^i_{b}}{\sum_{i=1}^n P^i_{b} Q^i_{b}} \times 100 \\ \\
       & = \sf \sum_{i=1}^n \frac{P_{t}^i}{P_{b}^i} \times w^i_b \times 100 
  \end{aligned}$$`
]

<br>
.font130[
&nbsp; &nbsp; &nbsp; `$\sf P^i_{t}$` &mdash; price of good `$i$` in year `$t$` <br>
&nbsp; &nbsp; &nbsp; `$\sf P^i_{b}$` &mdash; price of good `$i$` in the base year <br>
&nbsp; &nbsp; &nbsp; `$\sf Q^i_{b}$` &mdash; quantity of good `$i$` in the base year <br>
&nbsp; &nbsp; &nbsp; `$\sf w^i_b$` &mdash; weight of good `$i$` in the base year basket
]

---

# CPI Methodology Bulgaria

---

# Sub-Indices and Expenditures

---

# Sub-Indices and Weights

---

# CPI by Commodity Groups (1995 = 100)

---

# Other Price Indices
<br>

.font130[
<table class="none">
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b>core CPI</b></td>
    <td> </td>
    <td> </td>
    <td>a measure of the overall cost of consumer goods and services excluding food and energy.</td>
  </tr>
  <tr>
  <td> </td>
  <td> </td>
  <td> </td>
  <td> </td>
  </tr>
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b>producer price index (PPI)</b></td>
    <td> </td>
    <td> </td>
    <td>a measure of the cost of a basket of goods and services bought by firms.</td>
  </tr>
  <tr>
  <td> </td>
  <td> </td>
  <td> </td>
  <td> </td>
  </tr>
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b>GDP deflator </b></td>
    <td> </td>
    <td> </td>
    <td> a measure of the price level calculated as the ratio of nominal GDP to real GDP times 100.  </td>
  </tr>
</table>  
]

---

<hr>

---

# Three Problems with CPI

<br>

Three problems with the index are widely acknowledged but difficult to solve:

&nbsp; &nbsp; &nbsp; 1. Substitution bias;

&nbsp; &nbsp; &nbsp; 2. Introduction of new goods;

&nbsp; &nbsp; &nbsp; 3. Unmeasured quality change.

Let's look at the details of each problem!
]

---

# Substitution Bias

<br>

.font130[
Consumers respond to these differing price changes by buying less of the goods
whose prices have risen by relatively large amounts and by buying more of the
goods whose prices have risen less or perhaps even have fallen. That is, consumers
**substitute** toward goods that have become relatively less expensive.
]

.font130[
If a price
index is computed assuming a fixed basket of goods, it ignores the possibility of
consumer substitution and, therefore, **overstates the increase in the cost of living**
from one year to the next.
]

---

# Introduction of New Goods

<br>

.font130[
When
a new good is introduced, consumers have more variety from which to choose,
and this increased variety in turn reduces the cost of maintaining the same level of
economic well-being. 
]

.font130[
But because the CPI is based on a fixed basket of goods and services, it does
not reflect the increase in the value of the currency that results from the introduction
of new goods.
]

.font130[
If a price
index is computed assuming a fixed basket of goods, it ignores the benefit of variety and, therefore, **overstates the increase in the cost of living**
from one year to the next.
]

---

# Unmeasured Quality Change

.font130[
If the quality of a
good deteriorates from one year to the next while its price remains the same, you
are getting a lesser good for the same amount of money, so the value of the currency
falls.
]

.font130[
Similarly, if the quality rises from one year to the next, the value of the currency rises.
The BLS does its best to account for quality change. When the quality of a good in
the basket changes &mdash; for example, when a car model has more horsepower or gets
better gas mileage from one year to the next &mdash; the Bureau adjusts the price of the
good to account for the quality change. In doing so, it is trying to compute the price
of a **basket of goods of constant quality**.]

.font130[
Despite these efforts, changes in quality
remain a problem because quality is hard to measure, hence the CPI **overstates the increase in the cost of living** from one year to the next.
]

---

# Magnitude of Measurement Issues

<br>

.font130[
There is much debate among economists about how severe these measurement
problems are and what should be done about them. Studies put the upward bias
in measured inflation at about 0.5 to 1.0 percent per year.]

.font130[
The issue is important
because many government programs use the CPI to adjust for changes in the overall
level of prices. Recipients of Social Security, for instance, get annual increases in
benefits that are tied to the CPI.]

.font130[
Some economists have suggested modifying these
programs to correct for the measurement problems by, for instance, reducing the
magnitude of the automatic benefit increases.
]

---

# Jakob Fugger (1459 - 1525)

---

# Michael Jay Boskin

---

# The GDP Deflator and the CPI

.font130[
Measuring changes in the cost of living by using the GDP deflator has some serious limitations: 
]

.font130[
&nbsp; &nbsp; &nbsp; 1. The GDP deflator reflects the prices of ALL goods and services produced domestically,  including goods and services that are not directly consumed by the average household (e.g. investment goods, military equipment, etc.)
]

.font130[
&nbsp; &nbsp; &nbsp; 2. The GDP deflator reflects the prices of goods and services produced domestically, excluding imported goods and services even though they might be an important part of the consumption of the average household (e.g. Japanese electronics, Bangladeshi textiles, etc.)
]

.font130[
&nbsp; &nbsp; &nbsp; 3. The data for calculating the GDP deflator becomes available rather slowly, compared to the CPI which is available on monthly basis. 
]

---

# CPI Weights Versus GDP Deflator Weights

<br>
.font130[

`$$\begin{aligned}
\sf \text{CPI}_t  & = \sf \frac{\sum_{i=1}^n P^i_{t} Q^i_{b}}{\sum_{i=1}^n P^i_{b} Q^i_{b}} \times 100 \\ \\
\sf \text{GDP Deflator}_t  & = \sf \frac{\sum_{i=1}^m P^i_{t} Q^i_{t}}{\sum_{i=1}^m P^i_{b} Q^i_{t}} \times 100 
\end{aligned}$$`
<br>
&nbsp; &nbsp; &nbsp; `$\sf P^i_{t}$` &mdash; price of good `$i$` in year `$t$` <br>
&nbsp; &nbsp; &nbsp; `$\sf P^i_{b}$` &mdash; price of good `$i$` in the base year <br>
&nbsp; &nbsp; &nbsp; `$\sf Q^i_{b}$` &mdash; quantity of good `$i$` in the base year <br>
&nbsp; &nbsp; &nbsp; `$\sf Q^i_{t}$` &mdash; quantity of good `$i$` in year `$t$` <br>

]

---

# Bulgaria's GDP Deflator and CPI Inflation

<br>

---

# Bulgaria's GDP Deflator and CPI Inflation

<br>

---

<hr>

---

# Dollar Figures from Different Times

<br>

.font130[
The formula for converting dollar figures from year T into today’s dollars is the
following:
]

$$  \text{Amount in today’s dollars} = \text{Amount in year T dollars} \times \frac{\text{Price level today}}{\text{Price level in year T}} $$
--

.font130[
A price index such as the CPI or the GDP deflator measures the price level and thus determines the size
of the inflation correction:
]

`$$\begin{aligned} 
\sf \text{Amount in today’s dollars} & = \sf \text{Amount in year T dollars} \times \frac{\text{CPI today}}{\text{CPI in year T}} \\ \\
\sf \text{Amount in today’s dollars} & = \sf \text{Amount in year T dollars} \times \frac{\text{GDP deflator today}}{\text{GDP deflator in year T}}
\end{aligned}$$`

---

# Correcting Nominal GDP for Inflation

<br>

`$$\begin{aligned} 
\sf \text{Amount in today’s dollars} & = \sf \text{Amount in year T dollars} \times \frac{\text{GDP deflator today}}{\text{GDP deflator in year T}} \\ \\
\sf \text{Nominal GDP} & = \sf \text{Amount in base year dollars} \times \frac{\text{GDP deflator today}}{\text{GDP deflator in base year}} \\ \\
\sf \text{Nominal GDP} & = \sf \text{Real GDP} \times \frac{\text{GDP deflator today}}{\text{100}} \\ \\
\sf   \text{GDP deflator} & = \sf \frac{\text{Nominal GDP}}{\text{Real GDP}} \times {\text{100}}
\end{aligned}$$`

---

# Correcting Wages for Inflation

.font130[
In year 2000, the average wage in Bulgaria was 224 BGN and the CPI was 3,447. In year 2022, the average wage in Bulgaria was 1,739 BGN and the CPI was 8,516. Has the purchasing power of wages in Bulgaria increased or decreased in the 2000 &mdash; 2022 period? 
]

`$$\begin{aligned} 
\sf \text{Wage expressed in 2022 BGN} & = \sf \text{Wage expressed in 2000 BGN} \times \frac{\text{CPI in 2022}}{\text{CPI in 2000}} \\ \\
\sf \text{1,739 BGN} & = \sf \text{Wage expressed in 2000 BGN} \times \frac{\text{8,516}}{\text{3,447}} \\ \\
\sf \text{Wage expressed in 2000 BGN} & = \sf \frac{1,739 \times 3,447}{8,516} \\ \\
\sf \text{Wage expressed in 2000 BGN} & = \sf \text{705 BGN}
\end{aligned}$$`

--
.font130[
**Answer:** The 1,739 BGN paid in 2022 are equivalent to 705 BGN in 2000. <br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  This is more than 3 times the average wage in 2000! 
]

---

# Indexation

.font130[
<table class="none">
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b> indexation </b></td>
    <td> </td>
    <td> </td>
    <td>the automatic correction by law or contract of a dollar amount for the effects of inflation</td>
  </tr>
</table>  
]

--
.font130[
For example, some long-term contracts between firms and unions include
partial or complete indexation of the wage to the CPI. Such a provision, called a
*cost-of-living allowance* (or COLA), automatically raises the wage when the CPI
rises.
]

.font130[
Indexation is also a feature of many laws. Social Security benefits, for instance,
are adjusted every year to compensate the elderly for increases in prices. The brackets
of the federal income tax—the income levels at which the tax rates change—are
also indexed for inflation.
]
--

.font130[
There are, however, many ways in which the tax system
is not indexed for inflation, even when perhaps it should be!
]

---

<hr>

---

# Introduction to Interest Rates

.font130[
A major characteristics of loans is the difference in time
between taking the loan and paying it back with interest.
]

.font130[
The purchasing power of money may change between the
two moments and this needs to be accounted for to
calculate the actual interest on the loan.
]

.font130[
Recognizing the difference in purchasing power of money
in different moments leads to the recognition of two types
of interest rates: nominal and real.
]

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# Definitions

<br>

.font130[
<table class="none">
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b> nominal interest rate </b></td>
    <td> </td>
    <td> </td>
    <td>the interest rate as
usually reported without a
correction for the effects
of inflation</td>
  </tr>
</table>

The nominal interest rate simply shows by what percentage the
nominal sum of money has increased between the two moments.
]

.font130[
<table class="none">
  <tr>
    <td style="text-align: center; vertical-align: middle;"><b> real interest rate </b></td>
    <td> </td>
    <td> </td>
    <td>the interest rate corrected
for the effects of inflation</td>
  </tr>
</table>

The real interest rate tries to show by what percentage the real
purchasing power of the nominal sums of money has increased
between the two moments.
]

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# Real Interest Rate Formula

<br>

.font130[
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  **Real interest rate = Nominal interest rate - Inflation rate**
]

<br> 
--

.font130[
For example, if the nominal interest rate is 5% and the inflation rate is 3%, the real interest rate is 2% which means that the purchasing power of the bank deposit has increased:

`$$\sf 2\% = 5\% - 3\%$$`
]
--

.font130[
However, if the nominal interest rate is 5% and the inflation rate is 7%, the real interest rate is -2% which means that the purchasing power of the bank deposit has decreased:

`$$\sf -2\% = 5\% - 7\%$$`
]

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# Questions?

<br>

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# Thank You!

<br>

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