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Inflation: Why a Sandwich Costs More Than $5

In 2008 you could famously buy a foot-long sub for $5. In 2020, that same sub costs $6.75. As we look at generating enough passive income to cover our future expenses, we must not just think of how many dollars we will have, but how many sandwiches those dollars can buy!
Two Sandwiches

The difference between those two prices is the difference between a nominal price and a real price. Similarly, if we hide $100 under the mattress today, a decade from now we will still have $100 nominal dollars, but will have fewer real dollars, because we can buy less sandwiches with the same money.
Graph showing the cost of sandwiches

The green line shows the average price of sandwiches rising over time, with the shaded area projecting 30 years into the future based on the same growth rate. The blue line shows how the number of sandwiches we can buy for the same nominal money decreases as a result.

While sandwiches are an illustrative example, this effect touches all goods and services and is known as inflation. In the US, Consumer Price Index (CPI) is a measure tracked by the Bureau of Labor Statistics as measure of generally how much more expensive life is getting.

Graph showing Consumer Price Index

This measure of CPI from the Federal Reserve shows the signature characteristic curve of exponential growth.

To calculate the price of something after inflation, we must know the starting price, the rate of inflation, and the time period. For example, if we start with a $5.00 footlong and inflate it at 2.53% for 12 years, we get the following: $$Starting\ Price * (1 + Inflation\ Rate)^\text{Number of Years} = Final\ Price$$ $$$5.00 * (1.0253)^{12} = $6.75$$
So, what is happening in this equation? We start with the initial price of the sandwich. Then we multiply by the inflation rate. Notice that when we say something is inflating by 2.53%, what we really mean is that it will be 102.53% of the original price. 100% from the original price plus a 2.53% increase. Converting the percentages to decimal form: $$100\% = 1$$ $$2.53\% = 0.0253$$ $$100\%+2.53\%=1+0.0253=1.0253$$
The next step is to use an exponent: we raise 1.0253 to the power of 12, the number of periodsIn this case, the number of years. we are growing over. This calculation gives a growth of 2.53% per year, every year, for 12 years. It is exactly equivalent to just multiplying 1.025 by itself 12 times: $$1.0253^{12}=1.0253*1.0253*1.0253*1.0253*1.0253*1.0253*1.0253*1.0253*1.0253*1.0253*1.0253*1.0253$$ It is not enough to just add 2.53% per year, as each year’s inflation must start with the previous year’s already inflated price. This is a perfect example of exponential growthIn fact, we'll use the same exponential math to calculate our money after compounding investment returns..

CPI has historically averaged 2-3% per year and going forward the US Federal Reserve targets a 2% annual inflation rate. The Fed aren’t omniscient, but they do have some very powerful tools to help push towards this rate. I wouldn’t expect anything drastically different, and this ~2% will work to decay our spending power over time.

Since our spending power is constantly decaying, we must not hide our savings under a mattress. Even savings accounts at banks these days offer interest rates that are well under inflation and should only be utilized to keep cash for short term needs. Instead we invest our savings in stocks, bonds, real estate, or other assets that all work to create a return on our money greater than the rate of the rate of inflation to make sure we can continue to buy sandwiches well into the future.
Many Sandwiches

How did I know 2.53% was the right amount of inflation to grow from $5 to $6.75 over 12 years? If we start with the final price, initial price, and time period we can rework the equation to find the inflation rate: $$Starting\ Price * (1 + Inflation\ Rate)^\text{Number of Years} = Final\ Price$$
First, divide both sides of the equation by the Starting Price:

$${Starting\ Price * (1 + Inflation\ Rate)^\text{Number of Years} \over Starting\ Price} = {Final\ Price \over Starting\ Price}$$ $${(1 + Inflation\ Rate)^\text{Number of Years}} = {Final\ Price \over Starting\ Price}$$
Next, raise both sides to the power of \(\text{(1 / Number of Years)}\):

$${(1 + Inflation\ Rate)^\text{Number of Years}}^\text{(1 / Number of Years)} = \left( {Final\ Price \over Starting\ Price} \right)^\text{(1 / Number of Years)}$$
When raising a power to another power, you multiply the exponents together to get the resulting power. In this case: $$\text{Number of Years} * (1 / \text{Number of Years}) = 1$$ Which simplifies our equation to: $$(1 + Inflation\ Rate) = \left( {Final\ Price \over Starting\ Price} \right)^\text{(1 / Number of Years)}$$
Finally, subtract the 1 from both sides: $$Inflation\ Rate = \left( {Final\ Price \over Starting\ Price} \right)^\text{(1 / Number of Years)}-1$$ Now we can use the equation to calculate the inflation rate of our sandwich over the last 12 years: $$Inflation\ Rate = \left( {$6.75 \over $5.00} \right)^{ {\left( 1 \over 12 \right)} }-1 = 0.0253 = 2.53\%$$
Like the original form of this equation, this one is more generically useful as well. With a starting amount, ending amount, and number of time periods, we can calculate the rate of return for anything we are interested in, for example a stock or equity index fund. $$Growth\ Rate = \left( {Final\ Value \over Initial\ Value} \right)^\text{(1 / Number of Periods)}-1$$


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