Inflation, CPI and real value for HSC Maths Standard 2

Total change: $\frac{134.4 - 112.6}{112.6} \times 100\% = \frac{21.8}{112.6} \times 100\% \approx 19.4\%$.

Annual compound rate: $\left(\frac{134.4}{112.6}\right)^{1/5} - 1$.

$\frac{134.4}{112.6} \approx 1.1936$.

$(1.1936)^{1/5}$: take $\log$: $\frac{\log 1.1936}{5} = \frac{0.0769}{5} \approx 0.01537$, so $(1.1936)^{1/5} \approx 10^{0.01537} \approx 1.0360$.

Annual rate $\approx 3.6\%$ per year.

Markers reward the percentage-increase formula, and the geometric (compound) mean for the annual rate.

Inflation-adjusted (real) salary in 2024 dollars:

$\text{salary}_{2024} = 65000 \times \frac{137.8}{116.2} = 65000 \times 1.1859 \approx 77082.62$, i.e. $77082.62.

Markers reward the ratio of CPIs in the correct order (new over old) and the answer rounded to cents.

Use the percentage change in the CPI. The annual inflation rate is just the percentage change in the index over one year, with the earlier value on the bottom.

Evaluate.

State the answer. The annual inflation rate is about $3.3\%$. Check: the index rose by $3.4$ points on a base of $102$, and $\tfrac{3.4}{102}$ is a little above $3\%$, so $3.3\%$ is the right size.

Choose the right ratio. Total inflation is the change in the index expressed as a percentage of the earlier (base) value, so the earlier figure $108.6$ goes on the bottom.

Evaluate.

State the answer. Prices rose by about $14.6\%$ in total over the six years. Check: the index rose by $15.9$ points on a base near $109$, and $\tfrac{16}{109}$ is a little under $15\%$, so $14.6\%$ is the right size. This is total inflation, not an annual rate.

Set up the conversion. To restate money in a later year's dollars, multiply by the ratio of the new CPI to the old CPI, $\frac{CPI_{\text{new}}}{CPI_{\text{old}}}$, with the later year on top.

Evaluate the ratio, then the amount. The ratio is $\frac{132.0}{110.0} = 1.2$.

State the answer. The salary is worth $54000.00 in the later year's dollars. Check: the index rose by exactly one fifth ($110$ to $132$ is $+20\%$), so the figure should rise by one fifth too, and $45000 + 9000 = 54000$. The figure is larger because more dollars are needed to buy the same basket after prices rise.

Apply one year of inflation. A price that keeps pace with inflation is multiplied by $(1 + \text{rate})$, so multiply by $1.045$.

Evaluate.

State the answer. The pass is expected to cost $83.60 next year. Check: $4.5\%$ of $80 is $3.60, and $80 + 3.60 = 83.60$, so adding one year of inflation gives the same figure as multiplying by $1.045$.

Write the CPI ratio. The total multiplier over the six years is the ratio of the two index values.

Undo six years of compounding with a sixth root. The annual rate is the geometric mean, found by raising the ratio to the power $\tfrac{1}{6}$, never by dividing the total change by $6$.

Use logarithms for the fractional power.

State the answer. The equivalent annual inflation rate is about $2.5\%$ per year. Check: compounding $2.5\%$ for six years gives $(1.025)^6 \approx 1.16$, which returns the ratio $\frac{137.5}{118.8}$, so the rate is consistent. The total rise was about $15.7\%$, but the annual rate is only $2.5\%$, not $\tfrac{15.7}{6} \approx 2.6\%$, because inflation compounds.

Use the real return formula. Dividing the nominal growth factor by the inflation factor strips out the loss of purchasing power.

Evaluate.

State the answer. The real return is about $2.6\%$. Check: the quick approximation $r - i = 6.5 - 3.8 = 2.7\%$ is close to the exact $2.6\%$, which is the expected pattern (the exact value is slightly below the difference). At Standard 2 level the approximation $2.7\%$ is also accepted.

Choose compound, not simple, inflation. Inflation multiplies each year, so the bill is multiplied by $(1 + \text{rate})^n$ with rate $0.04$ and $n = 5$. Adding $4\% \times 5 = 20\%$ would understate the rise.

Evaluate the growth factor, then the amount.

State the answer. The expected weekly bill after five years is about $304.16. Check: the total rise is about $21.7\%$, a little more than the $20\%$ a simple sum would give, which is the expected effect of compounding. The figure is larger than the $250 start, as it must be when prices rise.

Find the percentage rise in the rent.

Find the percentage rise in the CPI (general inflation).

Compare. The rent rose about $33.3\%$ while prices in general rose about $23.2\%$, so the rent rose by more than inflation: in real terms the tenant is worse off.

Express the old rent in the new year's dollars. Multiply the old rent by $\frac{CPI_{\text{new}}}{CPI_{\text{old}}} = \frac{138.0}{112.0}$.

So $420 of old rent is equivalent to $517.50 in the new year. The actual new rent of $560 exceeds this by

State the answer. The rent outpaced inflation; in real terms it is about $42.50 per week higher than an inflation-only increase would have produced. Check: an inflation-matched rent would have been about $517.50, below the actual $560, confirming the rent rose faster than prices.

Decide the direction first. We are moving money back to an earlier, cheaper year, so the equivalent figure should be smaller. The CPI ratio must therefore be less than $1$, with the earlier (smaller) index on top.

Evaluate the ratio, then the amount. The ratio is $\frac{116.2}{137.8} \approx 0.8433$.

State the answer. The salary is worth about $80108.85 in the earlier year's dollars. Check: the figure is smaller than $95000, which is correct for moving back to a cheaper year; multiplying back by $\frac{137.8}{116.2}$ returns about $95000. Putting the larger CPI on top here would wrongly inflate the figure above $95000.

(a) Apply compound inflation. A fee that just keeps pace with inflation is multiplied by $(1 + \text{rate})^n$ with rate $0.035$ and $n = 7$.

Evaluate the growth factor, then the amount.

So an inflation-only fee would be about $22.90.

(b) Find the actual total percentage rise. Use the percentage change from $18 to $25.

Compare with inflation. Over the seven years inflation totalled $(1.035)^7 - 1 \approx 0.2723$, that is about $27.2\%$. The actual fee rose about $38.9\%$, well above the $27.2\%$ from inflation, so the fee rose faster than inflation.

State the answer. (a) The inflation-only fee is about $22.90. (b) The fee rose about $38.9\%$ in total, faster than the roughly $27.2\%$ caused by inflation. Check: the actual $25 sits above the inflation-only $22.90, confirming the fee outpaced inflation.

Find the nominal value after appreciation. Compound growth multiplies the price by $(1 + r)^n$ with $r = 0.055$ and $n = 8$.

So the painting is worth about $18416.24 in the later year's cash.

Convert that value back to purchase-year dollars. To compare purchasing power, restate the later amount in the earlier year's dollars by multiplying by $\frac{CPI_{\text{old}}}{CPI_{\text{new}}} = \frac{121.0}{149.0}$.

Compare with the purchase price. In purchase-year dollars the painting is worth about $14955.47, against the $12000 originally paid, a real gain of

about $24.6\%$ in real terms.

State the answer. The painting grew to about $18416.24 nominally, equal to about $14955.47 in purchase-year dollars, a real gain of roughly $2955.47. Check: the growth factor $1.5347$ exceeds the inflation factor $\frac{149.0}{121.0} \approx 1.2314$, so value rose faster than prices and the real gain is correctly positive.

Find the percentage rise in the salary.

Find the inflation rate over the same period.

Compare. The salary rose about $28.6\%$ while prices rose about $23.3\%$, so pay outpaced inflation and the worker is better off in real terms.

Quantify using the CPI ratio. Express the original salary in the later year's dollars by multiplying by $\frac{138.8}{112.6}$.

An inflation-matched salary would be about $69030.20, but the worker actually earns $72000, so the real gain is

State the answer. The worker is better off in real terms by about $2969.80 per year, because the $28.6\%$ pay rise exceeded the $23.3\%$ inflation. Check: the actual salary $72000 sits above the inflation-only figure of about $69030.20, confirming a real gain.