Purchasing power after inflation
Inflation raises prices while a fixed amount of money stays the same. Find the inflation factor for the whole span, then divide the fixed amount by it to express the future purchasing power in today's terms
\[\text{purchasing power}=\frac{A}{\left(1+\frac{r}{100}\right)^n}\]
- Build the inflation multiplier Add the annual inflation rate, written as a decimal, to 1 \(\displaystyle 1+\frac{4}{100}=1.04\)
- Raise the multiplier For n years, the yearly multiplier is raised to the power n \(\displaystyle F=1.04^5\)
- Divide fixed money Divide the fixed amount by the total inflation factor to find its purchasing power \(\displaystyle \$1{,}000\div1.04^5\approx\$822\)
- Check the direction With positive inflation, purchasing power must be less than the starting amount \(\displaystyle \$1{,}000\div1.04^5<\$1{,}000\)
Worked example
A fixed sum of $2,400 is kept without earning interest. Prices rise by average inflation of 3% per year for 8 years. Find the purchasing power of the sum after 8 years to the nearest dollar
\[\text{purchasing power}=\$2{,}400\div\left(1+\frac{3}{100}\right)^8\]
- \(\displaystyle R=1+\frac{3}{100}=1.03\)The average annual inflation rate gives a yearly price multiplier of 1.03
- \(\displaystyle R^8=1.03^8=1.2667700813876161\)Eight years of inflation use the yearly multiplier eight times
- \(\displaystyle \$2{,}400\div1.2667700813876161=\$1{,}894.58\ldots\approx\$1{,}895\)Divide the fixed sum by the full factor and round the result to the nearest dollar
Common mistakes
Multiplying instead of dividing
✕\(\displaystyle A\times\left(1+\frac{r}{100}\right)^n\)✓\(\displaystyle A\div\left(1+\frac{r}{100}\right)^n\)
The amount of money is fixed while prices rise, so the amount buys less and must be divided by the growth factorUsing the rate as the multiplier
✕\(\displaystyle A\div\left(0.04\right)^5\)✓\(\displaystyle A\div\left(1.04\right)^5\)
The yearly price factor includes the original 100% as well as the 4% increaseUsing a linear factor
✕\(\displaystyle A\div\left(1+5\times0.04\right)\)✓\(\displaystyle A\div\left(1.04\right)^5\)
Inflation is applied repeatedly, so the yearly factor is powered rather than increased by a linear totalDifficulty levels
introFind purchasing power after a moderate span using a whole-number inflation rate and nearest-dollar rounding
standardUse a decimal inflation rate over a longer span and sometimes find the purchasing power lost
challengeCompare purchasing power with a rising price or interpret the effect over a long span
examChoose the purchasing power produced by dividing by the powered inflation factor
Questions about this topic
What does purchasing power mean?+
Purchasing power describes how much a fixed sum can buy. Here, its value after inflation is expressed in today's dollars
Why is purchasing power divided by the inflation factor?+
Inflation makes the same goods more expensive. Dividing by the price-growth factor converts the fixed sum to the smaller amount of today's buying power it represents later
Why is the factor raised to the number of years?+
The inflation multiplier acts once in each year. Repeating the same multiplication for n years is written as the nth power
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 3: Sequences and finance
In this chapter
Identify the behaviour of a sequenceFind the rule for a sequence and give the next term3A Sequence patterns and rules: Generate the terms of a sequence from a starting value and a rule3B Terms and recurrence relations: Generate the terms of a sequence from a starting value and a ruleName and number the terms of a sequence3B Terms and recurrence relations: Write a recurrence relation for a sequenceFind the common difference and the next term of an arithmetic sequenceDecide whether a sequence is arithmetic, geometric or neitherTabulate a sequence and graph its terms3D Arithmetic recurrence and terms: Write a recurrence relation for a sequenceFind term n of an arithmetic sequence using the rule3D Arithmetic recurrence and terms: Find when a sequence first reaches or passes a given valueModel a simple-interest investment or loan with a recurrence relationModel flat-rate depreciation with a recurrence relationModel unit-cost depreciation with a recurrence relationUse the rule for term n of a linear growth or decay modelRead the financial parameters out of a linear recurrence relation3E Linear growth and decay in finance: Find when a sequence first reaches or passes a given valueBuild and compare linear financial modelsFind the common ratio and the next term of a geometric sequenceTabulate and interpret discrete geometric graphsDistinguish arithmetic and geometric patternsWrite and interpret geometric recurrence rulesWrite an explicit geometric sequence ruleFind term n of a geometric sequence using the ruleFind the first geometric threshold index3H Compound interest and depreciation: Find when a sequence first reaches or passes a given valueModel a compound-interest investment or loan with a recurrence relationWrite a recurrence relation for a loan or investment with a given compounding periodConstruct complete compound-interest recurrence modelsFind the first financial threshold periodConstruct and use reducing-balance depreciation models3I Geometric models and finance: Find when a sequence first reaches or passes a given valueUse the rule for term n of a compound-interest or reducing-balance modelFind the interest on a credit-card debt compounding dailyCompare the total cost of ways of paying, or the return on ways of investingFind the effect of inflation on a priceFind the purchasing power of an amount after inflation